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Для наглядности возьмём множество из трёх цифр: {1, 2, 3}. Формулы комбинаторики определяют количества возможных комбинаций этих элементов между собой.",[414,415,417],"h6",{"id":416},"перестановки","Перестановки",[402,419,420,421,424,425,427],{},"Берутся ",[405,422,423],{},"все"," ",[409,426,411],{}," элеметов исходного множества, меняется лишь порядок их следования друг за другом. В нашем случае — составляются \"слова\" из всех трёх элементов исходного множества: {1,2,3}, {1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}, {3,2,1}.",[402,429,430],{},"Количество неповторяющихся вариантов перестановки в этом случае определяется формулой:",[432,433,435],"h5",{"id":434},"",[436,437,441],"mjx-container",{"className":438,"jax":440},[439],"MathJax","SVG",[442,443,452,480],"svg",{"style":444,"xmlns":445,"width":446,"height":447,"role":448,"focusable":449,"viewBox":450,"xmlnsXLink":451},"vertical-align: -0.439ex;","http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","9.82ex","2.059ex","img","false","0 -716 4340.6 910","http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxlink",[453,454,455,460,464,468,472,476],"defs",{},[456,457],"path",{"id":458,"d":459},"MJX-1-TEX-I-1D443","M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z",[456,461],{"id":462,"d":463},"MJX-1-TEX-N-5F","M0 -62V-25H499V-62H0Z",[456,465],{"id":466,"d":467},"MJX-1-TEX-I-1D45B","M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z",[456,469],{"id":470,"d":471},"MJX-1-TEX-N-3D","M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z",[456,473],{"id":474,"d":475},"MJX-1-TEX-N-21","M78 661Q78 682 96 699T138 716T180 700T199 661Q199 654 179 432T158 206Q156 198 139 198Q121 198 119 206Q118 209 98 431T78 661ZM79 61Q79 89 97 105T141 121Q164 119 181 104T198 61Q198 31 181 16T139 1Q114 1 97 16T79 61Z",[456,477],{"id":478,"d":479},"MJX-1-TEX-N-2C","M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z",[481,482,486],"g",{"stroke":483,"fill":483,"stroke-width":484,"transform":485},"currentColor","0","scale(1,-1)",[481,487,489,497,504,511,519,524,531],{"dataMmlNode":488},"math",[481,490,492],{"dataMmlNode":491},"mi",[493,494],"use",{"dataC":495,"xLinkHref":496},"1D443","#MJX-1-TEX-I-1D443",[481,498,500],{"dataMmlNode":491,"transform":499},"translate(751,0)",[493,501],{"dataC":502,"xLinkHref":503},"5F","#MJX-1-TEX-N-5F",[481,505,507],{"dataMmlNode":491,"transform":506},"translate(1251,0)",[493,508],{"dataC":509,"xLinkHref":510},"1D45B","#MJX-1-TEX-I-1D45B",[481,512,515],{"dataMmlNode":513,"transform":514},"mo","translate(2128.8,0)",[493,516],{"dataC":517,"xLinkHref":518},"3D","#MJX-1-TEX-N-3D",[481,520,522],{"dataMmlNode":491,"transform":521},"translate(3184.6,0)",[493,523],{"dataC":509,"xLinkHref":510},[481,525,527],{"dataMmlNode":513,"transform":526},"translate(3784.6,0)",[493,528],{"dataC":529,"xLinkHref":530},"21","#MJX-1-TEX-N-21",[481,532,534],{"dataMmlNode":513,"transform":533},"translate(4062.6,0)",[493,535],{"dataC":536,"xLinkHref":537},"2C","#MJX-1-TEX-N-2C",[402,539,540],{},"где ! — обозначение факториала (n! = 1 ∙ 2 ∙ 3 ∙ ... ∙ n). В нашем примере оно равно 3! = 6.",[414,542,544],{"id":543},"размещения","Размещения",[402,546,547,548,550],{},"Из исходного множества (\"алфавита\" из n элементов) берётся только m каких-то элементов, и выполняются различные перестановки только этого количества элементов. При этом рассматриваются ",[405,549,423],{}," возможные способы выборки такого количества элементов из исходного множества.",[402,552,553],{},"Например, когда из исходного множества из n = 3 цифр: {1, 2, 3} берутся подмножества из m = 2 цифр: {1,2}, {1,3}, и {2,3}, получаем следующую подборку комбинаций:\n{1,2}, {2,1}, {1,3}, {3,1}, {2,3}, {3,2}.",[402,555,556],{},"Количество различных возможных вариантов таких комбинаций элементов вычисляется по формуле:",[432,558,560],{"id":559},"_1",[436,561,563],{"className":562,"jax":440},[439],[442,564,569,603],{"style":565,"xmlns":445,"width":566,"height":567,"role":448,"focusable":449,"viewBox":568,"xmlnsXLink":451},"vertical-align: -1.238ex;","15.089ex","3.275ex","0 -900.3 6669.3 1447.4",[453,570,571,575,578,581,585,588,591,595,599],{},[456,572],{"id":573,"d":574},"MJX-2-TEX-I-1D434","M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",[456,576],{"id":577,"d":463},"MJX-2-TEX-N-5F",[456,579],{"id":580,"d":467},"MJX-2-TEX-I-1D45B",[456,582],{"id":583,"d":584},"MJX-2-TEX-I-1D45A","M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z",[456,586],{"id":587,"d":471},"MJX-2-TEX-N-3D",[456,589],{"id":590,"d":475},"MJX-2-TEX-N-21",[456,592],{"id":593,"d":594},"MJX-2-TEX-N-28","M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z",[456,596],{"id":597,"d":598},"MJX-2-TEX-N-2212","M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z",[456,600],{"id":601,"d":602},"MJX-2-TEX-N-29","M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z",[481,604,605],{"stroke":483,"fill":483,"stroke-width":484,"transform":485},[481,606,607,613,619,635,641],{"dataMmlNode":488},[481,608,609],{"dataMmlNode":491},[493,610],{"dataC":611,"xLinkHref":612},"1D434","#MJX-2-TEX-I-1D434",[481,614,616],{"dataMmlNode":491,"transform":615},"translate(750,0)",[493,617],{"dataC":502,"xLinkHref":618},"#MJX-2-TEX-N-5F",[481,620,623,628],{"dataMmlNode":621,"transform":622},"msup","translate(1250,0)",[481,624,625],{"dataMmlNode":491},[493,626],{"dataC":509,"xLinkHref":627},"#MJX-2-TEX-I-1D45B",[481,629,631],{"dataMmlNode":491,"transform":630},"translate(633,363) scale(0.707)",[493,632],{"dataC":633,"xLinkHref":634},"1D45A","#MJX-2-TEX-I-1D45A",[481,636,638],{"dataMmlNode":513,"transform":637},"translate(2831.6,0)",[493,639],{"dataC":517,"xLinkHref":640},"#MJX-2-TEX-N-3D",[481,642,646],{"dataMmlNode":643,"dataMjxTexclass":644,"transform":645},"TeXAtom","ORD","translate(3887.4,0)",[481,647,649,663,701],{"dataMmlNode":648},"mfrac",[481,650,653,657],{"dataMmlNode":651,"transform":652},"mrow","translate(1080.5,394) scale(0.707)",[481,654,655],{"dataMmlNode":491},[493,656],{"dataC":509,"xLinkHref":627},[481,658,660],{"dataMmlNode":513,"transform":659},"translate(600,0)",[493,661],{"dataC":529,"xLinkHref":662},"#MJX-2-TEX-N-21",[481,664,666,672,677,684,689,696],{"dataMmlNode":651,"transform":665},"translate(220,-370.3) scale(0.707)",[481,667,668],{"dataMmlNode":513},[493,669],{"dataC":670,"xLinkHref":671},"28","#MJX-2-TEX-N-28",[481,673,675],{"dataMmlNode":491,"transform":674},"translate(389,0)",[493,676],{"dataC":509,"xLinkHref":627},[481,678,680],{"dataMmlNode":513,"transform":679},"translate(989,0)",[493,681],{"dataC":682,"xLinkHref":683},"2212","#MJX-2-TEX-N-2212",[481,685,687],{"dataMmlNode":491,"transform":686},"translate(1767,0)",[493,688],{"dataC":633,"xLinkHref":634},[481,690,692],{"dataMmlNode":513,"transform":691},"translate(2645,0)",[493,693],{"dataC":694,"xLinkHref":695},"29","#MJX-2-TEX-N-29",[481,697,699],{"dataMmlNode":513,"transform":698},"translate(3034,0)",[493,700],{"dataC":529,"xLinkHref":662},[702,703],"rect",{"width":704,"height":705,"x":706,"y":707},2541.9,60,"120","220",[402,709,710],{},"В нашем случае оно равно: 3! \u002F (3 - 2)! = 3! \u002F 1! = 6.",[414,712,714],{"id":713},"сочетания","Сочетания",[402,716,717],{},"Из исходного множества n берётся m каких-то элементов, но в пределах каждого такого способа выборки никакие перестановки не производятся, т.е. нас интересуют не все возможные комбинации перестановок, а только лишь количество возможных подмножеств(порядок следования элементов в каждом подмножестве не важен).",[402,719,720],{},"Например, из исходного множества из n = 3 цифр: {1,2,3} берутся подмножества из m = 2 цифр: {1,2}, {1,3}, {2,3}. Варианты {2,1}, {3,1} и {3,2} считаются совпадающими с предыдущими.",[402,722,723],{},"Количество возможных таких сочетаний определяется по формуле:",[432,725,727],{"id":726},"_2",[436,728,730],{"className":729,"jax":440},[439],[442,731,734,768],{"style":565,"xmlns":445,"width":732,"height":567,"role":448,"focusable":449,"viewBox":733,"xmlnsXLink":451},"17.406ex","0 -900.3 7693.3 1447.4",[453,735,736,740,743,746,749,752,755,758,761,764],{},[456,737],{"id":738,"d":739},"MJX-3-TEX-I-1D436","M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z",[456,741],{"id":742,"d":463},"MJX-3-TEX-N-5F",[456,744],{"id":745,"d":467},"MJX-3-TEX-I-1D45B",[456,747],{"id":748,"d":584},"MJX-3-TEX-I-1D45A",[456,750],{"id":751,"d":471},"MJX-3-TEX-N-3D",[456,753],{"id":754,"d":475},"MJX-3-TEX-N-21",[456,756],{"id":757,"d":594},"MJX-3-TEX-N-28",[456,759],{"id":760,"d":598},"MJX-3-TEX-N-2212",[456,762],{"id":763,"d":602},"MJX-3-TEX-N-29",[456,765],{"id":766,"d":767},"MJX-3-TEX-N-22C5","M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z",[481,769,770],{"stroke":483,"fill":483,"stroke-width":484,"transform":485},[481,771,772,778,784,797,803],{"dataMmlNode":488},[481,773,774],{"dataMmlNode":491},[493,775],{"dataC":776,"xLinkHref":777},"1D436","#MJX-3-TEX-I-1D436",[481,779,781],{"dataMmlNode":491,"transform":780},"translate(760,0)",[493,782],{"dataC":502,"xLinkHref":783},"#MJX-3-TEX-N-5F",[481,785,787,792],{"dataMmlNode":621,"transform":786},"translate(1260,0)",[481,788,789],{"dataMmlNode":491},[493,790],{"dataC":509,"xLinkHref":791},"#MJX-3-TEX-I-1D45B",[481,793,794],{"dataMmlNode":491,"transform":630},[493,795],{"dataC":633,"xLinkHref":796},"#MJX-3-TEX-I-1D45A",[481,798,800],{"dataMmlNode":513,"transform":799},"translate(2841.6,0)",[493,801],{"dataC":517,"xLinkHref":802},"#MJX-3-TEX-N-3D",[481,804,806],{"dataMmlNode":643,"dataMjxTexclass":644,"transform":805},"translate(3897.4,0)",[481,807,808,820,866],{"dataMmlNode":648},[481,809,811,815],{"dataMmlNode":651,"transform":810},"translate(1587.5,394) scale(0.707)",[481,812,813],{"dataMmlNode":491},[493,814],{"dataC":509,"xLinkHref":791},[481,816,817],{"dataMmlNode":513,"transform":659},[493,818],{"dataC":529,"xLinkHref":819},"#MJX-3-TEX-N-21",[481,821,822,827,831,836,840,845,849,856,861],{"dataMmlNode":651,"transform":665},[481,823,824],{"dataMmlNode":513},[493,825],{"dataC":670,"xLinkHref":826},"#MJX-3-TEX-N-28",[481,828,829],{"dataMmlNode":491,"transform":674},[493,830],{"dataC":509,"xLinkHref":791},[481,832,833],{"dataMmlNode":513,"transform":679},[493,834],{"dataC":682,"xLinkHref":835},"#MJX-3-TEX-N-2212",[481,837,838],{"dataMmlNode":491,"transform":686},[493,839],{"dataC":633,"xLinkHref":796},[481,841,842],{"dataMmlNode":513,"transform":691},[493,843],{"dataC":694,"xLinkHref":844},"#MJX-3-TEX-N-29",[481,846,847],{"dataMmlNode":513,"transform":698},[493,848],{"dataC":529,"xLinkHref":819},[481,850,852],{"dataMmlNode":513,"transform":851},"translate(3312,0)",[493,853],{"dataC":854,"xLinkHref":855},"22C5","#MJX-3-TEX-N-22C5",[481,857,859],{"dataMmlNode":491,"transform":858},"translate(3590,0)",[493,860],{"dataC":633,"xLinkHref":796},[481,862,864],{"dataMmlNode":513,"transform":863},"translate(4468,0)",[493,865],{"dataC":529,"xLinkHref":819},[702,867],{"width":868,"height":705,"x":706,"y":707},3555.9,[402,870,871],{},"В нашем случае оно равно: 3! \u002F (3 - 2)! ∙ 2! = 3! \u002F 2! = 3.",[414,873,875],{"id":874},"перестановки-с-повторениями","Перестановки с повторениями",[402,877,878],{},"Пусть исходное множество может, кроме уникальных, неповторяющихся, содержать какие-то одинаковые элементы, например: {1,1,2,2,2,3,3,3,3}.",[402,880,881],{},"Тогда при всех возможных перестановках будут возникать одинаковые варианты, в которых местами меняются только одинаковые элементы. Такие повторы необходимо исключить при подсчёте количества неповторяющихся комбинаций.",[402,883,884],{},"Формула количества возможных перестановок с повторениями:",[432,886,888],{"id":887},"_3",[436,889,891],{"className":890,"jax":440},[439],[442,892,897,946],{"style":893,"xmlns":445,"width":894,"height":895,"role":448,"focusable":449,"viewBox":896,"xmlnsXLink":451},"vertical-align: -0.883ex;","36.759ex","3.145ex","0 -1000 16247.7 1390.1",[453,898,899,902,906,909,912,915,919,922,926,930,934,937,940,943],{},[456,900],{"id":901,"d":459},"MJX-4-TEX-I-1D443",[456,903],{"id":904,"d":905},"MJX-4-TEX-S4-2013","M0 248V285H499V248H0Z",[456,907],{"id":908,"d":594},"MJX-4-TEX-N-28",[456,910],{"id":911,"d":467},"MJX-4-TEX-I-1D45B",[456,913],{"id":914,"d":463},"MJX-4-TEX-N-5F",[456,916],{"id":917,"d":918},"MJX-4-TEX-N-31","M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",[456,920],{"id":921,"d":479},"MJX-4-TEX-N-2C",[456,923],{"id":924,"d":925},"MJX-4-TEX-N-32","M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z",[456,927],{"id":928,"d":929},"MJX-4-TEX-N-2E","M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z",[456,931],{"id":932,"d":933},"MJX-4-TEX-I-1D458","M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z",[456,935],{"id":936,"d":602},"MJX-4-TEX-N-29",[456,938],{"id":939,"d":471},"MJX-4-TEX-N-3D",[456,941],{"id":942,"d":475},"MJX-4-TEX-N-21",[456,944],{"id":945,"d":767},"MJX-4-TEX-N-22C5",[481,947,948],{"stroke":483,"fill":483,"stroke-width":484,"transform":485},[481,949,950,972,977,983,989,997,1003,1008,1013,1020,1025,1032,1037,1042,1047,1052,1057,1064,1070,1076,1188],{"dataMmlNode":488},[481,951,953,958],{"dataMmlNode":952},"mover",[481,954,955],{"dataMmlNode":491},[493,956],{"dataC":495,"xLinkHref":957},"#MJX-4-TEX-I-1D443",[481,959,961],{"dataMmlNode":513,"transform":960},"translate(0,615)",[442,962,967],{"width":963,"height":964,"x":484,"y":965,"viewBox":966},751,237,"148","187.8 148 751 237",[493,968],{"dataC":969,"xLinkHref":970,"transform":971},"2013","#MJX-4-TEX-S4-2013","scale(2.253,1)",[481,973,974],{"dataMmlNode":513,"transform":499},[493,975],{"dataC":670,"xLinkHref":976},"#MJX-4-TEX-N-28",[481,978,980],{"dataMmlNode":491,"transform":979},"translate(1140,0)",[493,981],{"dataC":509,"xLinkHref":982},"#MJX-4-TEX-I-1D45B",[481,984,986],{"dataMmlNode":491,"transform":985},"translate(1740,0)",[493,987],{"dataC":502,"xLinkHref":988},"#MJX-4-TEX-N-5F",[481,990,993],{"dataMmlNode":991,"transform":992},"mn","translate(2240,0)",[493,994],{"dataC":995,"xLinkHref":996},"31","#MJX-4-TEX-N-31",[481,998,1000],{"dataMmlNode":513,"transform":999},"translate(2740,0)",[493,1001],{"dataC":536,"xLinkHref":1002},"#MJX-4-TEX-N-2C",[481,1004,1006],{"dataMmlNode":491,"transform":1005},"translate(3184.7,0)",[493,1007],{"dataC":509,"xLinkHref":982},[481,1009,1011],{"dataMmlNode":491,"transform":1010},"translate(3784.7,0)",[493,1012],{"dataC":502,"xLinkHref":988},[481,1014,1016],{"dataMmlNode":991,"transform":1015},"translate(4284.7,0)",[493,1017],{"dataC":1018,"xLinkHref":1019},"32","#MJX-4-TEX-N-32",[481,1021,1023],{"dataMmlNode":513,"transform":1022},"translate(4784.7,0)",[493,1024],{"dataC":536,"xLinkHref":1002},[481,1026,1028],{"dataMmlNode":513,"transform":1027},"translate(5229.3,0)",[493,1029],{"dataC":1030,"xLinkHref":1031},"2E","#MJX-4-TEX-N-2E",[481,1033,1035],{"dataMmlNode":513,"transform":1034},"translate(5674,0)",[493,1036],{"dataC":1030,"xLinkHref":1031},[481,1038,1040],{"dataMmlNode":513,"transform":1039},"translate(6118.7,0)",[493,1041],{"dataC":1030,"xLinkHref":1031},[481,1043,1045],{"dataMmlNode":513,"transform":1044},"translate(6563.3,0)",[493,1046],{"dataC":536,"xLinkHref":1002},[481,1048,1050],{"dataMmlNode":491,"transform":1049},"translate(7008,0)",[493,1051],{"dataC":509,"xLinkHref":982},[481,1053,1055],{"dataMmlNode":491,"transform":1054},"translate(7608,0)",[493,1056],{"dataC":502,"xLinkHref":988},[481,1058,1060],{"dataMmlNode":491,"transform":1059},"translate(8108,0)",[493,1061],{"dataC":1062,"xLinkHref":1063},"1D458","#MJX-4-TEX-I-1D458",[481,1065,1067],{"dataMmlNode":513,"transform":1066},"translate(8629,0)",[493,1068],{"dataC":694,"xLinkHref":1069},"#MJX-4-TEX-N-29",[481,1071,1073],{"dataMmlNode":513,"transform":1072},"translate(9295.8,0)",[493,1074],{"dataC":517,"xLinkHref":1075},"#MJX-4-TEX-N-3D",[481,1077,1079],{"dataMmlNode":643,"dataMjxTexclass":644,"transform":1078},"translate(10351.6,0)",[481,1080,1081,1093,1185],{"dataMmlNode":648},[481,1082,1084,1088],{"dataMmlNode":651,"transform":1083},"translate(2498.7,394) scale(0.707)",[481,1085,1086],{"dataMmlNode":491},[493,1087],{"dataC":509,"xLinkHref":982},[481,1089,1090],{"dataMmlNode":513,"transform":659},[493,1091],{"dataC":529,"xLinkHref":1092},"#MJX-4-TEX-N-21",[481,1094,1096,1100,1104,1109,1114,1120,1125,1130,1135,1140,1145,1150,1155,1160,1165,1170,1175,1180],{"dataMmlNode":651,"transform":1095},"translate(220,-346.3) scale(0.707)",[481,1097,1098],{"dataMmlNode":491},[493,1099],{"dataC":509,"xLinkHref":982},[481,1101,1102],{"dataMmlNode":491,"transform":659},[493,1103],{"dataC":502,"xLinkHref":988},[481,1105,1107],{"dataMmlNode":991,"transform":1106},"translate(1100,0)",[493,1108],{"dataC":995,"xLinkHref":996},[481,1110,1112],{"dataMmlNode":513,"transform":1111},"translate(1600,0)",[493,1113],{"dataC":529,"xLinkHref":1092},[481,1115,1117],{"dataMmlNode":513,"transform":1116},"translate(1878,0)",[493,1118],{"dataC":854,"xLinkHref":1119},"#MJX-4-TEX-N-22C5",[481,1121,1123],{"dataMmlNode":491,"transform":1122},"translate(2156,0)",[493,1124],{"dataC":509,"xLinkHref":982},[481,1126,1128],{"dataMmlNode":491,"transform":1127},"translate(2756,0)",[493,1129],{"dataC":502,"xLinkHref":988},[481,1131,1133],{"dataMmlNode":991,"transform":1132},"translate(3256,0)",[493,1134],{"dataC":1018,"xLinkHref":1019},[481,1136,1138],{"dataMmlNode":513,"transform":1137},"translate(3756,0)",[493,1139],{"dataC":529,"xLinkHref":1092},[481,1141,1143],{"dataMmlNode":513,"transform":1142},"translate(4034,0)",[493,1144],{"dataC":854,"xLinkHref":1119},[481,1146,1148],{"dataMmlNode":513,"transform":1147},"translate(4312,0)",[493,1149],{"dataC":1030,"xLinkHref":1031},[481,1151,1153],{"dataMmlNode":513,"transform":1152},"translate(4590,0)",[493,1154],{"dataC":1030,"xLinkHref":1031},[481,1156,1158],{"dataMmlNode":513,"transform":1157},"translate(4868,0)",[493,1159],{"dataC":1030,"xLinkHref":1031},[481,1161,1163],{"dataMmlNode":513,"transform":1162},"translate(5146,0)",[493,1164],{"dataC":854,"xLinkHref":1119},[481,1166,1168],{"dataMmlNode":491,"transform":1167},"translate(5424,0)",[493,1169],{"dataC":509,"xLinkHref":982},[481,1171,1173],{"dataMmlNode":491,"transform":1172},"translate(6024,0)",[493,1174],{"dataC":502,"xLinkHref":988},[481,1176,1178],{"dataMmlNode":491,"transform":1177},"translate(6524,0)",[493,1179],{"dataC":1062,"xLinkHref":1063},[481,1181,1183],{"dataMmlNode":513,"transform":1182},"translate(7045,0)",[493,1184],{"dataC":529,"xLinkHref":1092},[702,1186],{"width":1187,"height":705,"x":706,"y":707},5378.1,[481,1189,1191],{"dataMmlNode":513,"transform":1190},"translate(15969.7,0)",[493,1192],{"dataC":536,"xLinkHref":1002},[402,1194,1195],{},"где n = n1 + n2 + ... + nk.",[402,1197,1198],{},"Здесь n — общее количество элементов в исходном множестве (для нашего примера — 9), а n1, n2, n3 ... — количества элементов в каждой группе из одинаковых элементов(в примере имеется n1 = 2 элемента \"1\", n2 = 3 элемента \"2\", n3 = 4 элемента \"3\").",[1200,1201,1202],"blockquote",{},[402,1203,1204],{},"Если в исходном множестве есть неповторяющиеся (уникальные) элементы, то каждый из них считается входящим в группу из одного такого элемента. Далее можно применить ту же формулу. Однако, учитывая, что 1! = 1 и фактически уникальные элементы на результат расчётов никак не влияют, можно принять за основу следующее правило: в указанной формуле просто брать значение n равным общему количеству элементов множества (как уникальных, так и повторяющихся), а в качестве n1, n2, n3 ... брать численности групп из нескольких повторяющихся элементов.",[402,1206,1207],{},"В итоге количество перестановок с повторениями в нашем примере равно: 9! \u002F 2! ∙ 3! ∙ 4! = 1260.",[414,1209,1211],{"id":1210},"размещения-с-повторениями","Размещения с повторениями",[402,1213,1214,1215,1218],{},"Пусть имеется исходное множество из ",[409,1216,1217],{},"различных"," элементов, из которого требуется выбрать k элементов и расставлять их на n местах, так что каждый элемент может использоваться несколько раз, а может не использоваться совсем.",[402,1220,1221],{},"Либо: имеется исходное множество из k элементов, при этом можно брать их все любое количество раз, расставляя в n знакоместах.",[402,1223,1224],{},"Формула:",[432,1226,1228],{"id":1227},"_4",[436,1229,1231],{"className":1230,"jax":440},[439],[442,1232,1237,1260],{"style":1233,"xmlns":445,"width":1234,"height":1235,"role":448,"focusable":449,"viewBox":1236,"xmlnsXLink":451},"vertical-align: -0.186ex;","11.179ex","2.667ex","0 -1096.7 4941.2 1178.7",[453,1238,1239,1242,1245,1248,1251,1254,1257],{},[456,1240],{"id":1241,"d":574},"MJX-5-TEX-I-1D434",[456,1243],{"id":1244,"d":463},"MJX-5-TEX-N-5F",[456,1246],{"id":1247,"d":467},"MJX-5-TEX-I-1D45B",[456,1249],{"id":1250,"d":933},"MJX-5-TEX-I-1D458",[456,1252],{"id":1253,"d":905},"MJX-5-TEX-S4-2013",[456,1255],{"id":1256,"d":471},"MJX-5-TEX-N-3D",[456,1258],{"id":1259,"d":929},"MJX-5-TEX-N-2E",[481,1261,1262],{"stroke":483,"fill":483,"stroke-width":484,"transform":485},[481,1263,1264,1302,1308,1320],{"dataMmlNode":488},[481,1265,1266,1291],{"dataMmlNode":952},[481,1267,1268,1273,1278],{"dataMmlNode":651},[481,1269,1270],{"dataMmlNode":491},[493,1271],{"dataC":611,"xLinkHref":1272},"#MJX-5-TEX-I-1D434",[481,1274,1275],{"dataMmlNode":491,"transform":615},[493,1276],{"dataC":502,"xLinkHref":1277},"#MJX-5-TEX-N-5F",[481,1279,1280,1285],{"dataMmlNode":621,"transform":622},[481,1281,1282],{"dataMmlNode":491},[493,1283],{"dataC":509,"xLinkHref":1284},"#MJX-5-TEX-I-1D45B",[481,1286,1288],{"dataMmlNode":491,"transform":1287},"translate(633,289) scale(0.707)",[493,1289],{"dataC":1062,"xLinkHref":1290},"#MJX-5-TEX-I-1D458",[481,1292,1294],{"dataMmlNode":513,"transform":1293},"translate(0,711.7)",[442,1295,1298],{"width":1296,"height":964,"x":484,"y":965,"viewBox":1297},2301.4,"575.4 148 2301.4 237",[493,1299],{"dataC":969,"xLinkHref":1300,"transform":1301},"#MJX-5-TEX-S4-2013","scale(6.904,1)",[481,1303,1305],{"dataMmlNode":513,"transform":1304},"translate(2579.2,0)",[493,1306],{"dataC":517,"xLinkHref":1307},"#MJX-5-TEX-N-3D",[481,1309,1311,1315],{"dataMmlNode":621,"transform":1310},"translate(3635,0)",[481,1312,1313],{"dataMmlNode":491},[493,1314],{"dataC":1062,"xLinkHref":1290},[481,1316,1318],{"dataMmlNode":491,"transform":1317},"translate(554,363) scale(0.707)",[493,1319],{"dataC":509,"xLinkHref":1284},[481,1321,1323],{"dataMmlNode":513,"transform":1322},"translate(4663.2,0)",[493,1324],{"dataC":1030,"xLinkHref":1325},"#MJX-5-TEX-N-2E",[402,1327,1328],{},"Фактически же в данном случае можно считать, что рассматривается количество всех возможных n-значных чисел в системе счисления с основанием k.",[402,1330,1331,1334],{},[409,1332,1333],{},"Пример:"," {1,2,3,4,5,6,7} — имеются 4 нечётных и 3 чётных цифры. Требуется определить количество различных (неповторяющихся) комбинаций из трёх нечётных цифр. Получается: k = 4 (всего нечётных цифр в исходном множестве), n = 3 (берём любые три цифры из этого набора и расставляем на трёх позициях). Тогда количество возможных комбинаций равно 4^3 = 64.",[414,1336,1338],{"id":1337},"сочетания-с-повторениями","Сочетания с повторениями",[402,1340,1341],{},"Имеется множество, в котором, возможно, есть одинаковые элементы n различных разновидностей. Из него берётся некоторое количество k элементов (как одной, так и разных разновидностей; возможно, что какой-то вид элементов в тех или иных комбинациях вообще не будет выбран). Требуется узнать количество возможных различных комбинаций (наборов) таких элементов.",[402,1343,1224],{},[432,1345,1347],{"id":1346},"_5",[436,1348,1350],{"className":1349,"jax":440},[439],[442,1351,1355,1400],{"style":565,"xmlns":445,"width":1352,"height":1353,"role":448,"focusable":449,"viewBox":1354,"xmlnsXLink":451},"32.849ex","3.719ex","0 -1096.7 14519.2 1643.8",[453,1356,1357,1360,1363,1366,1369,1372,1375,1379,1382,1385,1388,1391,1394,1397],{},[456,1358],{"id":1359,"d":739},"MJX-6-TEX-I-1D436",[456,1361],{"id":1362,"d":463},"MJX-6-TEX-N-5F",[456,1364],{"id":1365,"d":467},"MJX-6-TEX-I-1D45B",[456,1367],{"id":1368,"d":933},"MJX-6-TEX-I-1D458",[456,1370],{"id":1371,"d":905},"MJX-6-TEX-S4-2013",[456,1373],{"id":1374,"d":471},"MJX-6-TEX-N-3D",[456,1376],{"id":1377,"d":1378},"MJX-6-TEX-N-2B","M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z",[456,1380],{"id":1381,"d":598},"MJX-6-TEX-N-2212",[456,1383],{"id":1384,"d":918},"MJX-6-TEX-N-31",[456,1386],{"id":1387,"d":594},"MJX-6-TEX-N-28",[456,1389],{"id":1390,"d":602},"MJX-6-TEX-N-29",[456,1392],{"id":1393,"d":475},"MJX-6-TEX-N-21",[456,1395],{"id":1396,"d":767},"MJX-6-TEX-N-22C5",[456,1398],{"id":1399,"d":929},"MJX-6-TEX-N-2E",[481,1401,1402],{"stroke":483,"fill":483,"stroke-width":484,"transform":485},[481,1403,1404,1440,1446,1458,1463,1494,1499,1597],{"dataMmlNode":488},[481,1405,1406,1430],{"dataMmlNode":952},[481,1407,1408,1413,1418],{"dataMmlNode":651},[481,1409,1410],{"dataMmlNode":491},[493,1411],{"dataC":776,"xLinkHref":1412},"#MJX-6-TEX-I-1D436",[481,1414,1415],{"dataMmlNode":491,"transform":780},[493,1416],{"dataC":502,"xLinkHref":1417},"#MJX-6-TEX-N-5F",[481,1419,1420,1425],{"dataMmlNode":621,"transform":786},[481,1421,1422],{"dataMmlNode":491},[493,1423],{"dataC":509,"xLinkHref":1424},"#MJX-6-TEX-I-1D45B",[481,1426,1427],{"dataMmlNode":491,"transform":1287},[493,1428],{"dataC":1062,"xLinkHref":1429},"#MJX-6-TEX-I-1D458",[481,1431,1432],{"dataMmlNode":513,"transform":1293},[442,1433,1436],{"width":1434,"height":964,"x":484,"y":965,"viewBox":1435},2311.4,"577.9 148 2311.4 237",[493,1437],{"dataC":969,"xLinkHref":1438,"transform":1439},"#MJX-6-TEX-S4-2013","scale(6.934,1)",[481,1441,1443],{"dataMmlNode":513,"transform":1442},"translate(2589.2,0)",[493,1444],{"dataC":517,"xLinkHref":1445},"#MJX-6-TEX-N-3D",[481,1447,1449,1453],{"dataMmlNode":621,"transform":1448},"translate(3645,0)",[481,1450,1451],{"dataMmlNode":491},[493,1452],{"dataC":776,"xLinkHref":1412},[481,1454,1456],{"dataMmlNode":491,"transform":1455},"translate(845.3,363) scale(0.707)",[493,1457],{"dataC":1062,"xLinkHref":1429},[481,1459,1461],{"dataMmlNode":491,"transform":1460},"translate(4908.6,0)",[493,1462],{"dataC":502,"xLinkHref":1417},[481,1464,1466,1470,1477,1482,1488],{"dataMmlNode":643,"dataMjxTexclass":644,"transform":1465},"translate(5408.6,0)",[481,1467,1468],{"dataMmlNode":491},[493,1469],{"dataC":509,"xLinkHref":1424},[481,1471,1473],{"dataMmlNode":513,"transform":1472},"translate(822.2,0)",[493,1474],{"dataC":1475,"xLinkHref":1476},"2B","#MJX-6-TEX-N-2B",[481,1478,1480],{"dataMmlNode":491,"transform":1479},"translate(1822.4,0)",[493,1481],{"dataC":1062,"xLinkHref":1429},[481,1483,1485],{"dataMmlNode":513,"transform":1484},"translate(2565.7,0)",[493,1486],{"dataC":682,"xLinkHref":1487},"#MJX-6-TEX-N-2212",[481,1489,1491],{"dataMmlNode":991,"transform":1490},"translate(3565.9,0)",[493,1492],{"dataC":995,"xLinkHref":1493},"#MJX-6-TEX-N-31",[481,1495,1497],{"dataMmlNode":513,"transform":1496},"translate(9752.3,0)",[493,1498],{"dataC":517,"xLinkHref":1445},[481,1500,1502],{"dataMmlNode":643,"dataMjxTexclass":644,"transform":1501},"translate(10808.1,0)",[481,1503,1504,1546,1594],{"dataMmlNode":648},[481,1505,1507,1512,1516,1520,1524,1529,1534,1540],{"dataMmlNode":651,"transform":1506},"translate(220,516.8) 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scale(0.707)",[481,1550,1551],{"dataMmlNode":491},[493,1552],{"dataC":1062,"xLinkHref":1429},[481,1554,1556],{"dataMmlNode":513,"transform":1555},"translate(521,0)",[493,1557],{"dataC":529,"xLinkHref":1545},[481,1559,1561],{"dataMmlNode":513,"transform":1560},"translate(799,0)",[493,1562],{"dataC":854,"xLinkHref":1563},"#MJX-6-TEX-N-22C5",[481,1565,1567],{"dataMmlNode":513,"transform":1566},"translate(1077,0)",[493,1568],{"dataC":670,"xLinkHref":1511},[481,1570,1572],{"dataMmlNode":491,"transform":1571},"translate(1466,0)",[493,1573],{"dataC":509,"xLinkHref":1424},[481,1575,1577],{"dataMmlNode":513,"transform":1576},"translate(2066,0)",[493,1578],{"dataC":682,"xLinkHref":1487},[481,1580,1582],{"dataMmlNode":991,"transform":1581},"translate(2844,0)",[493,1583],{"dataC":995,"xLinkHref":1493},[481,1585,1587],{"dataMmlNode":513,"transform":1586},"translate(3344,0)",[493,1588],{"dataC":694,"xLinkHref":1539},[481,1590,1592],{"dataMmlNode":513,"transform":1591},"translate(3733,0)",[493,1593],{"dataC":529,"xLinkHref":1545},[702,1595],{"width":1596,"height":705,"x":706,"y":707},3193.2,[481,1598,1600],{"dataMmlNode":513,"transform":1599},"translate(14241.2,0)",[493,1601],{"dataC":1030,"xLinkHref":1602},"#MJX-6-TEX-N-2E",[402,1604,1605,1607],{},[409,1606,1333],{}," исходное множество {1,1,2,2,2,3,3,3,3}. Требуется найти количество различных наборов, получаемых из 5 любых выбранных из такого множества элементов. Здесь n = 3 (три возможных разновидности: \"1\", \"2\" и \"3\", при этом количества элементов каждой разновидности значения не имеют), а k = 5 (каждый набор состоит из 5 каких-то элементов.) Тогда количество различных наборов равно:",[432,1609,1611],{"id":1610},"_6",[436,1612,1614],{"className":1613,"jax":440},[439],[442,1615,1619,1672],{"style":565,"xmlns":445,"width":1616,"height":1617,"role":448,"focusable":449,"viewBox":1618,"xmlnsXLink":451},"44.314ex","3.674ex","0 -1076.9 19586.9 1624",[453,1620,1621,1624,1627,1631,1635,1638,1641,1644,1647,1650,1653,1656,1659,1662,1666,1669],{},[456,1622],{"id":1623,"d":739},"MJX-7-TEX-I-1D436",[456,1625],{"id":1626,"d":463},"MJX-7-TEX-N-5F",[456,1628],{"id":1629,"d":1630},"MJX-7-TEX-N-33","M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z",[456,1632],{"id":1633,"d":1634},"MJX-7-TEX-N-35","M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z",[456,1636],{"id":1637,"d":905},"MJX-7-TEX-S4-2013",[456,1639],{"id":1640,"d":471},"MJX-7-TEX-N-3D",[456,1642],{"id":1643,"d":1378},"MJX-7-TEX-N-2B",[456,1645],{"id":1646,"d":598},"MJX-7-TEX-N-2212",[456,1648],{"id":1649,"d":918},"MJX-7-TEX-N-31",[456,1651],{"id":1652,"d":594},"MJX-7-TEX-N-28",[456,1654],{"id":1655,"d":602},"MJX-7-TEX-N-29",[456,1657],{"id":1658,"d":475},"MJX-7-TEX-N-21",[456,1660],{"id":1661,"d":767},"MJX-7-TEX-N-22C5",[456,1663],{"id":1664,"d":1665},"MJX-7-TEX-N-37","M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z",[456,1667],{"id":1668,"d":925},"MJX-7-TEX-N-32",[456,1670],{"id":1671,"d":929},"MJX-7-TEX-N-2E",[481,1673,1674],{"stroke":483,"fill":483,"stroke-width":484,"transform":485},[481,1675,1676,1716,1722,1733,1738,1768,1773,1871,1876,1921,1926],{"dataMmlNode":488},[481,1677,1678,1705],{"dataMmlNode":952},[481,1679,1680,1685,1690],{"dataMmlNode":651},[481,1681,1682],{"dataMmlNode":491},[493,1683],{"dataC":776,"xLinkHref":1684},"#MJX-7-TEX-I-1D436",[481,1686,1687],{"dataMmlNode":491,"transform":780},[493,1688],{"dataC":502,"xLinkHref":1689},"#MJX-7-TEX-N-5F",[481,1691,1692,1698],{"dataMmlNode":621,"transform":786},[481,1693,1694],{"dataMmlNode":991},[493,1695],{"dataC":1696,"xLinkHref":1697},"33","#MJX-7-TEX-N-33",[481,1699,1701],{"dataMmlNode":991,"transform":1700},"translate(533,289) scale(0.707)",[493,1702],{"dataC":1703,"xLinkHref":1704},"35","#MJX-7-TEX-N-35",[481,1706,1708],{"dataMmlNode":513,"transform":1707},"translate(0,691.9)",[442,1709,1712],{"width":1710,"height":964,"x":484,"y":965,"viewBox":1711},2196.6,"549.1 148 2196.6 237",[493,1713],{"dataC":969,"xLinkHref":1714,"transform":1715},"#MJX-7-TEX-S4-2013","scale(6.59,1)",[481,1717,1719],{"dataMmlNode":513,"transform":1718},"translate(2474.3,0)",[493,1720],{"dataC":517,"xLinkHref":1721},"#MJX-7-TEX-N-3D",[481,1723,1725,1729],{"dataMmlNode":621,"transform":1724},"translate(3530.1,0)",[481,1726,1727],{"dataMmlNode":491},[493,1728],{"dataC":776,"xLinkHref":1684},[481,1730,1731],{"dataMmlNode":991,"transform":1455},[493,1732],{"dataC":1703,"xLinkHref":1704},[481,1734,1736],{"dataMmlNode":491,"transform":1735},"translate(4778.9,0)",[493,1737],{"dataC":502,"xLinkHref":1689},[481,1739,1741,1745,1751,1756,1762],{"dataMmlNode":643,"dataMjxTexclass":644,"transform":1740},"translate(5278.9,0)",[481,1742,1743],{"dataMmlNode":991},[493,1744],{"dataC":1696,"xLinkHref":1697},[481,1746,1748],{"dataMmlNode":513,"transform":1747},"translate(722.2,0)",[493,1749],{"dataC":1475,"xLinkHref":1750},"#MJX-7-TEX-N-2B",[481,1752,1754],{"dataMmlNode":991,"transform":1753},"translate(1722.4,0)",[493,1755],{"dataC":1703,"xLinkHref":1704},[481,1757,1759],{"dataMmlNode":513,"transform":1758},"translate(2444.7,0)",[493,1760],{"dataC":682,"xLinkHref":1761},"#MJX-7-TEX-N-2212",[481,1763,1765],{"dataMmlNode":991,"transform":1764},"translate(3444.9,0)",[493,1766],{"dataC":995,"xLinkHref":1767},"#MJX-7-TEX-N-31",[481,1769,1771],{"dataMmlNode":513,"transform":1770},"translate(9501.6,0)",[493,1772],{"dataC":517,"xLinkHref":1721},[481,1774,1776],{"dataMmlNode":643,"dataMjxTexclass":644,"transform":1775},"translate(10557.4,0)",[481,1777,1778,1821,1868],{"dataMmlNode":648},[481,1779,1780,1785,1789,1794,1799,1804,1809,1815],{"dataMmlNode":651,"transform":1506},[481,1781,1782],{"dataMmlNode":513},[493,1783],{"dataC":670,"xLinkHref":1784},"#MJX-7-TEX-N-28",[481,1786,1787],{"dataMmlNode":991,"transform":674},[493,1788],{"dataC":1696,"xLinkHref":1697},[481,1790,1792],{"dataMmlNode":513,"transform":1791},"translate(889,0)",[493,1793],{"dataC":1475,"xLinkHref":1750},[481,1795,1797],{"dataMmlNode":991,"transform":1796},"translate(1667,0)",[493,1798],{"dataC":1703,"xLinkHref":1704},[481,1800,1802],{"dataMmlNode":513,"transform":1801},"translate(2167,0)",[493,1803],{"dataC":682,"xLinkHref":1761},[481,1805,1807],{"dataMmlNode":991,"transform":1806},"translate(2945,0)",[493,1808],{"dataC":995,"xLinkHref":1767},[481,1810,1812],{"dataMmlNode":513,"transform":1811},"translate(3445,0)",[493,1813],{"dataC":694,"xLinkHref":1814},"#MJX-7-TEX-N-29",[481,1816,1818],{"dataMmlNode":513,"transform":1817},"translate(3834,0)",[493,1819],{"dataC":529,"xLinkHref":1820},"#MJX-7-TEX-N-21",[481,1822,1823,1827,1832,1838,1843,1848,1853,1858,1863],{"dataMmlNode":651,"transform":1548},[481,1824,1825],{"dataMmlNode":991},[493,1826],{"dataC":1703,"xLinkHref":1704},[481,1828,1830],{"dataMmlNode":513,"transform":1829},"translate(500,0)",[493,1831],{"dataC":529,"xLinkHref":1820},[481,1833,1835],{"dataMmlNode":513,"transform":1834},"translate(778,0)",[493,1836],{"dataC":854,"xLinkHref":1837},"#MJX-7-TEX-N-22C5",[481,1839,1841],{"dataMmlNode":513,"transform":1840},"translate(1056,0)",[493,1842],{"dataC":670,"xLinkHref":1784},[481,1844,1846],{"dataMmlNode":991,"transform":1845},"translate(1445,0)",[493,1847],{"dataC":1696,"xLinkHref":1697},[481,1849,1851],{"dataMmlNode":513,"transform":1850},"translate(1945,0)",[493,1852],{"dataC":682,"xLinkHref":1761},[481,1854,1856],{"dataMmlNode":991,"transform":1855},"translate(2723,0)",[493,1857],{"dataC":995,"xLinkHref":1767},[481,1859,1861],{"dataMmlNode":513,"transform":1860},"translate(3223,0)",[493,1862],{"dataC":694,"xLinkHref":1814},[481,1864,1866],{"dataMmlNode":513,"transform":1865},"translate(3612,0)",[493,1867],{"dataC":529,"xLinkHref":1820},[702,1869],{"width":1870,"height":705,"x":706,"y":707},3107.6,[481,1872,1874],{"dataMmlNode":513,"transform":1873},"translate(14182.8,0)",[493,1875],{"dataC":517,"xLinkHref":1721},[481,1877,1879],{"dataMmlNode":643,"dataMjxTexclass":644,"transform":1878},"translate(15238.5,0)",[481,1880,1881,1894,1918],{"dataMmlNode":648},[481,1882,1884,1890],{"dataMmlNode":651,"transform":1883},"translate(593.4,394) scale(0.707)",[481,1885,1886],{"dataMmlNode":991},[493,1887],{"dataC":1888,"xLinkHref":1889},"37","#MJX-7-TEX-N-37",[481,1891,1892],{"dataMmlNode":513,"transform":1829},[493,1893],{"dataC":529,"xLinkHref":1820},[481,1895,1896,1900,1904,1908,1913],{"dataMmlNode":651,"transform":1095},[481,1897,1898],{"dataMmlNode":991},[493,1899],{"dataC":1703,"xLinkHref":1704},[481,1901,1902],{"dataMmlNode":513,"transform":1829},[493,1903],{"dataC":529,"xLinkHref":1820},[481,1905,1906],{"dataMmlNode":513,"transform":1834},[493,1907],{"dataC":854,"xLinkHref":1837},[481,1909,1910],{"dataMmlNode":991,"transform":1840},[493,1911],{"dataC":1018,"xLinkHref":1912},"#MJX-7-TEX-N-32",[481,1914,1916],{"dataMmlNode":513,"transform":1915},"translate(1556,0)",[493,1917],{"dataC":529,"xLinkHref":1820},[702,1919],{"width":1920,"height":705,"x":706,"y":707},1496.8,[481,1922,1924],{"dataMmlNode":513,"transform":1923},"translate(17253.1,0)",[493,1925],{"dataC":517,"xLinkHref":1721},[481,1927,1929,1931,1933],{"dataMmlNode":991,"transform":1928},"translate(18308.9,0)",[493,1930],{"dataC":1018,"xLinkHref":1912},[493,1932],{"dataC":995,"xLinkHref":1767,"transform":1829},[493,1934],{"dataC":1030,"xLinkHref":1935,"transform":1936},"#MJX-7-TEX-N-2E","translate(1000,0)",[402,1938,1939],{},[409,1940,1941],{},"На этом основные формулы заканчиваются.",[1943,1944,1945],"style",{},"\nmjx-container[jax=\"SVG\"] {\n  direction: ltr;\n}\n\nmjx-container[jax=\"SVG\"] > svg {\n  overflow: visible;\n  min-height: 1px;\n  min-width: 1px;\n}\n\nmjx-container[jax=\"SVG\"] > svg a {\n  fill: blue;\n  stroke: blue;\n}\n\nmjx-container[jax=\"SVG\"][display=\"true\"] {\n  display: block;\n  text-align: center;\n  margin: 1em 0;\n}\n\nmjx-container[jax=\"SVG\"][display=\"true\"][width=\"full\"] {\n  display: flex;\n}\n\nmjx-container[jax=\"SVG\"][justify=\"left\"] {\n  text-align: left;\n}\n\nmjx-container[jax=\"SVG\"][justify=\"right\"] {\n  text-align: right;\n}\n\ng[data-mml-node=\"merror\"] > g {\n  fill: red;\n  stroke: red;\n}\n\ng[data-mml-node=\"merror\"] > rect[data-background] {\n  fill: yellow;\n  stroke: none;\n}\n\ng[data-mml-node=\"mtable\"] > line[data-line], svg[data-table] > g > line[data-line] {\n  stroke-width: 70px;\n  fill: none;\n}\n\ng[data-mml-node=\"mtable\"] > rect[data-frame], svg[data-table] > g > rect[data-frame] {\n  stroke-width: 70px;\n  fill: none;\n}\n\ng[data-mml-node=\"mtable\"] > .mjx-dashed, svg[data-table] > g > .mjx-dashed {\n  stroke-dasharray: 140;\n}\n\ng[data-mml-node=\"mtable\"] > .mjx-dotted, svg[data-table] > g > .mjx-dotted {\n  stroke-linecap: round;\n  stroke-dasharray: 0,140;\n}\n\ng[data-mml-node=\"mtable\"] > g > svg {\n  overflow: visible;\n}\n\n[jax=\"SVG\"] mjx-tool {\n  display: inline-block;\n  position: relative;\n  width: 0;\n  height: 0;\n}\n\n[jax=\"SVG\"] mjx-tool > mjx-tip {\n  position: absolute;\n  top: 0;\n  left: 0;\n}\n\nmjx-tool > mjx-tip {\n  display: inline-block;\n  padding: .2em;\n  border: 1px solid #888;\n  font-size: 70%;\n  background-color: #F8F8F8;\n  color: black;\n  box-shadow: 2px 2px 5px #AAAAAA;\n}\n\ng[data-mml-node=\"maction\"][data-toggle] {\n  cursor: pointer;\n}\n\nmjx-status {\n  display: block;\n  position: fixed;\n  left: 1em;\n  bottom: 1em;\n  min-width: 25%;\n  padding: .2em .4em;\n  border: 1px solid #888;\n  font-size: 90%;\n  background-color: #F8F8F8;\n  color: black;\n}\n\nforeignObject[data-mjx-xml] {\n  font-family: initial;\n  line-height: normal;\n  overflow: visible;\n}\n\nmjx-container[jax=\"SVG\"] path[data-c], mjx-container[jax=\"SVG\"] use[data-c] {\n  stroke-width: 3;\n}\n",{"title":434,"searchDepth":1947,"depth":1948,"links":1949},2,5,[1950],{"id":399,"depth":1951,"text":400,"children":1952},3,[1953,1955,1958,1961,1964,1967,1970,1971],{"id":416,"depth":1954,"text":417},6,{"id":434,"depth":1948,"text":434,"children":1956},[1957],{"id":543,"depth":1954,"text":544},{"id":559,"depth":1948,"text":434,"children":1959},[1960],{"id":713,"depth":1954,"text":714},{"id":726,"depth":1948,"text":434,"children":1962},[1963],{"id":874,"depth":1954,"text":875},{"id":887,"depth":1948,"text":434,"children":1965},[1966],{"id":1210,"depth":1954,"text":1211},{"id":1227,"depth":1948,"text":434,"children":1968},[1969],{"id":1337,"depth":1954,"text":1338},{"id":1346,"depth":1948,"text":434},{"id":1610,"depth":1948,"text":434},"2024-10-17","Основные формулы комбинаторики.","md","images\u002Fblog\u002Ftoi\u002Fst2\u002Fimg.png",{},8,true,{"title":355,"description":1973},"mKI73gMoZl8F0S_Gj6iutvpAEHBoQv78xSDDdwwLGRc",[1982,1984],{"title":335,"path":336,"stem":337,"description":1983,"children":-1},"Информация и информационные процессы. Кодирование информации.",{"title":359,"path":360,"stem":361,"description":1985,"children":-1},"Адрес документа в Интернете",1780737506405]