Веленкин;Комбинаторика
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-sjkj 9,, иcЗПииcЛТЛФc fММТЛМ‘cЛК‰
·néno I( ) I(N) ësnuntz I(N + ) ² ·néno I( ) I(N + ) q
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Ívs•oy¹x• énryééntzt€u xvvztv¡ntqnu q tj·js•t€uq ·sn tjuq uv tv vlqt oj léymqu k€wqx€kjz• ·snt€ wvxsnlvkjzns• tvxzq wéq·nu éjtv qsq wvoltv u€ wvsy·qu síivp nn ·snt Ìltjrv wéq ëzvu tju wéqlnzx¹ k€wqxjz• q kxn wénl€ly•qn ·snt€ ² knl• tn yotjk q} u€ tn yotjnu q wvxsnlyí•q} ·sntvk Ëv kv utvmq} xsy·j¹} u€ }vzqu yotjz• zvs•rv vlqt vwénlnsntt€p ·snt wvxsnlvkjzns•tvxzq j vxzjs•t€n ·snt€ tju tn ty t€ Æ ëzq} xsy·j¹} ylvitnn qunz• ¹ktyí {véuysy ls¹ Q mv ·sntj wvxsnlvkjzns•tvxzq ΀ iylnu mvkvéqz• ·zv tnrv zvéj¹ wvxsnlvkjzns•tvxz• ¹ks¹nzx¹ én¡ntqnu ljttvmv énryé éntztvmv xvvztv¡ntq¹ nxsq wéq wvlxzjtvkrn ëzvp wvxsnlvkj zns•tvxzq xvvztv¡ntqn zv lnxzknttv k€wvst¹nzx¹ Ëjwéquné
wvxsnlvkjzns•tvxz• Q ¹ks¹nzx¹ vltqu qo én¡ntqp énryééntztvmv xvvztv¡ntq¹ I(Q + ) = I(Q + ) − I(Q)
Æ xjuvu lnsn nxsq I(Q) = Q zv
I(Q + ) − I(Q) = Q+ − Q = ( − ) Q+ = Q+ = I(Q + )
wéq síivu Q Ívëzvuy I(Q) = Q ¹ks¹nzx¹ én¡ntqnu yrjojttvmv
xvvztv¡ntq¹
èn¡ntqn énryééntztvmv xvvztv¡ntq¹ N mv wvé¹lrj tjo€kj nzx¹ vi•qu nxsq vtv ojkqxqz vz N wévqokvs•t€} wvxzv¹tt€} & &N q wyznu wvlivéj ëzq} wvxzv¹tt€} wvokvs¹nz wvsy
·qz• síivn én¡ntqn ljttvmv xvvztv¡ntq¹ Ëjwéquné ls¹ xvvztv¡ntq¹
I(Q + ) = I(Q + ) − I(Q) |
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vi•qu én¡ntqnu iylnz |
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I(Q) = & Q + & Q |
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Æ xjuvu lnsn snmrv wévknéqz• ·zv wvxsnlvkjzns•tvxz• vi éj•jnz xvvztv¡ntqn k zv lnxzkv Ívëzvuy tju tjlv zvs•rv wvrjojz• ·zv síivn én¡ntqn ëzvmv xvvztv¡ntq¹ uv tv wénl xzjkqz• k kqln Ëv síivn én¡ntqn xvvztv¡ntq¹ vltv otj·tv vwénlns¹nzx¹ otj·ntq¹uq I( ) q I( ) Ívëzvuy tju tjlv
lvrjojz• ·zv ls¹ síi€} ·qxns D q E tjplyzx¹ zjrqn otj·ntq¹ & q & ·zv & + & = D q & + & = E Ëv snmrv kqlnz• ·zv wéq síi€} otj·ntq¹} D q E xqxznuj yéjktntqp
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& + & = D |
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& + & = E |
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fsy·jp wvxzv¹tt€} rvë{{q|qntzvk |
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qunnz én¡ntqn Ívëzvuy lnpxzkqzns•tv ¹ks¹nzx¹ vi•qu én ¡ntqnu xvvztv¡ntq¹
fsy·jp wvxzv¹tt€} rvë{{q|qntzvk
bs¹ én¡ntq¹ énryééntzt€} xvvztv¡ntqp vi•q} wéjkqs kvvi•n mvkvé¹ tnz Ìltjrv xy•nxzkynz knx•uj ·jxzv kxzén·jí •qpx¹ rsjxx xvvztv¡ntqp én¡jnu€p nlqtvviéjot€u unzvlvu êzv ² énryééntzt€n xvvztv¡ntq¹ kqlj
I(Q + N) = D I(Q + N − ) + D I(Q + N − ) + + DN I(Q)
mln D |
D |
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² tnrvzvé€n ·qxsj Òjrqn xvvztv¡ntq¹ tj |
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N |
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o€kjíz sqtnpt€uq énryééntzt€uq xvvztv¡ntq¹uq x wvxzv¹t |
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t€uq rvë{{q|qntzjuq |
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ftj·jsj u€ éjxxuvzéqu rjr én¡jízx¹ zjrqn xvvztv¡ntq¹ |
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wéq N = z n qoy·qu xvvztv¡ntq¹ kqlj |
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I(Q + ) = D I(Q + ) + D I(Q) |
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èn¡ntqn ëzq} xvvztv¡ntqp vxtvkjtv tj xsnlyí•q} yzkné lntq¹}
cxsq I (Q) q I (Q) ¹ks¹ízx¹ én¡ntq¹uq énryééntztvmv xvvztv¡ntq¹ zv wéq síi€} ·qxsj} $ q % wvxsnlvkjzns•
tvxz• |
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I(Q) = $I (Q) + % I (Q) |
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zjr n ¹ks¹nzx¹ én¡ntqnu ëzvmv xvvztv¡ntq¹ |
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Æ xjuvu lnsn wv yxsvkqí qunnu |
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I (Q + ) = D I (Q + ) + D I (Q) |
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I (Q + ) = D I (Q + ) + D I (Q) |
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Ïutv qu ëzq éjkntxzkj tj $ q % xvvzknzxzknttv q xsv qu wvsy·ntt€n zv lnxzkj ΀ wvsy·qu ·zv
$ I (Q + ) + %I (Q + ) =
= D >$ I (Q + ) + %I (Q + )@ + D >$I (Q) + % I (Q)@ ¬ ëzv q votj·jnz ·zv $ I (Q) + %I (Q) ¹ks¹nzx¹ én¡ntqnu
xvvztv¡ntq¹
cxsq ·qxsv U ¹ks¹nzx¹ rvétnu rkjléjztvmv yéjktntq¹
U = D U + D |
zv wvxsnlvkjzns•tvxz• U |
U |
UQ− |
¹ks¹ |
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nzx¹ én¡ntqnu énryééntztvmv xvvztv¡ntq¹ I(Q + ) = D I(Q + ) + D I(Q)
-sjkj 9,, иcЗПииcЛТЛФc fММТЛМ‘cЛК‰
Æ |
xjuvu |
lnsn |
nxsq |
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I(Q) = UQ− |
zv |
I(Q + ) = UQ |
q |
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I(Q + |
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Ívlxzjks¹¹ |
ëzq |
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otj·ntq¹ |
k |
xvvztv¡ntqn |
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wvsy·jnu éjkntxzkv UQ+ = D |
UQ + D UQ− Ìtv xwéjknlsqkv zjr |
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rjr wv yxsvkqí qunnu U = D |
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U |
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+ D |
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Ëjé¹ly x wvxsnlvkjzns•tvxz•í ^UQ− ` síij¹ wvxsnlvkj |
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zns•tvxz• kqlj I(Q) |
= UQ+P |
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Q = |
zjr n |
¹ks¹nzx¹ |
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én¡ntqnu xvvztv¡ntq¹ |
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bs¹ lvrjojzns•xzkj lvxzjzv·tv qxwvs•ovkjz• yzkné |
lntqn |
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wvsv |
qk k $ = UP+ % = |
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Êo yzkné lntqp q k€znrjnz xsnlyí•nn wéjkqsv én¡n tq¹ sqtnpt€} énryééntzt€} xvvztv¡ntqp kzvévmv wvé¹lrj x wvxzv¹tt€uq rvë{{q|qntzjuq
Íyxz• ljtv énryééntztvn xvvztv¡ntqn |
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I(Q + ) = D I(Q + ) + D I(Q) |
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fvxzjkqu rkjléjztvn yéjktntqn |
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U = D U + D |
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rvzvévn tjo€kjnzx¹ }jéjrznéqxzq·nxrqu ls¹ ljttvmv xvvztv ¡ntq¹ cxsq ëzv yéjktntqn qunnz lkj éjosq·t€} rvét¹ U q
U zv vi•nn én¡ntqn xvvztv¡ntq¹ qunnz kql
I(Q) = & UQ− + & UQ− |
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ïzvi€ lvrjojz• ëzv wéjkqsv ojunzqu xtj·jsj ·zv wv yz kné lntqí én¡ntq¹uq tj¡nmv xvvztv¡ntq¹ ¹ks¹ízx¹
I |
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(Q) = UQ− |
q I |
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(Q) = UQ− ¬ zvmlj wv yzkné lntqí ¹ks¹nzx¹ |
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nmv én¡ntqnu q & UQ− + & UQ− Ëjlv zvs•rv wvrjojz• ·zv
síivn én¡ntqn xvvztv¡ntq¹ uv tv ojwqxjz• k ëzvu kqln
Ëv síivn én¡ntqn xvvztv¡ntq¹ kzvévmv wvé¹lrj vwénlns¹nzx¹ otj·ntq¹uq I( ) q I( ) Ívëzvuy lvxzjzv·tv wvrjojz• ·zv xq
xznuj yéjktntqp
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& + & = D |
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& U |
+ & U = E |
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fsy·jp wvxzv¹tt€} rvë{{q|qntzvk |
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qunnz én¡ntqn wéq síi€} D q E êzqu én¡ntqnu |
¹ks¹nzx¹ |
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& = |
E − DU |
& = |
DU − E |
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U − U |
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U − U |
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fsy·jp rvmlj vij rvét¹ yéjktntq¹ xvkwjljíz léym x léymvu u€ éjoinénu tnxrvs•rv wvo n ¬ xnp·jx wéqknlnu wéquné tj lvrjojttvn wéjkqsv
Íéq qoy·ntqq ·qxns Ðqivtj··q u€ wéq¡sq r énryééntz tvuy xvvztv¡ntqí
I(Q) = I(Q − ) + I(Q − ) |
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bs¹ tnmv }jéjrznéqxzq·nxrvn yéjktntqn qunnz kql |
U = U + |
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Çvét¹uq ëzvmv rkjléjztvmv yéjktntq¹ ¹ks¹ízx¹ ·qxsj |
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U = + √ |
U |
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= − √ |
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Ívëzvuy vi•nn én¡ntqn xvvztv¡ntq¹ Ðqivtj··q qunnz kql |
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+ √ Q |
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− √ Q |
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I(Q) = & |
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+ & |
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u€ kvxwvs•ovkjsqx• yzkné lntqnu q ko¹sq wvrjojznsq Q kunxzv Q −
΀ tjo€kjsq ·qxsjuq Ðqivtj··q én¡ntqn xvvztv¡ntq¹
ylvksnzkvé¹í•nn tj·js•t€u yxsvkq¹u I( ) = q I( ) =
z n wvxsnlvkjzns•tvxz• ïjxzv i€kjnz ylvitv lvijkqz• r ëzvp wvxsnlvkjzns•tvxzq ktj·jsn ·qxsv q éjxxujzéqkjz• wvxsnlvkjzns•tvxz•
‰xtv ·zv ëzj wvxsnlvkjzns•tvxz• ylvksnzkvé¹nz zvuy |
n xjuvuy |
énryééntztvuy xvvztv¡ntqí q tj·js•t€u |
yxsvkq¹u |
I( ) = I( ) = Ívsjmj¹ k {véuysn Q = q Q = wvsy·jnu |
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ls¹ & q & xqxznuy yéjktntqp
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& + & = |
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√ (& − & ) = |
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Ìzxílj tj}vlqu ·zv & = − & = √ q wvzvuy
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+ √ Q |
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− √ Q |
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I(Q) = |
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− |
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√ |
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Ëj wnék€p koms¹l rj nzx¹ ylqkqzns•t€u ·zv ëzv k€éj ntqn wéq kxn} tjzyéjs•t€} otj·ntq¹} Q wéqtqujnz |nsv·qxsntt€n otj·ntq¹ Ëv ëzv xzjtvkqzx¹ ivsnn wvt¹zt€u nxsq ojunzqz•
-sjkj 9,, иcЗПииcЛТЛФc fММТЛМ‘cЛК‰
·zv wvxsn éjxré€zq¹ kxn} xrvivr k€éj ntqn iylnz qunz• kql D + E√ tv wéq ojuntn √ tj −√ k€éj ntqn tn qountqzx¹
wvunt¹nz otjr q wnék€p xvutv qzns• q rkjléjztj¹ xrvirj wvëzvuy kxnmlj iylnz E =
fsy·jp éjkt€} rvétnp }jéjrznéqxzq·nxrvmv yéjktntq¹
Ìxzjtvkqux¹ znwné• tj xsy·jn rvmlj vij rvét¹ }jéjrznéqx |
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zq·nxrvmv yéjktntq¹ xvkwjljíz U |
= U Æ ëzvu xsy·jn k€éj n |
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tqn & |
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UQ− + & |
UQ− y n tn iylnz vi•qu én¡ntqnu Ænl• qo oj |
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zvmv ·zv U = U ëzv én¡ntqn uv |
tv ojwqxjz• k kqln |
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I(Q) = (& |
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+ & )UQ− = &UQ− |
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Ï tjx vxzjnzx¹ zvs•rv vltv wévqokvs•tvn wvxzv¹ttvn |
& q k€ |
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iéjz• nmv zjr ·zvi€ ylvksnzkvéqz• lkyu tj·js•t€u yxsvkq¹u |
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I( ) = D I( ) |
= E kvvi•n mvkvé¹ tnkvouv tv Ívëzvuy tju tjlv |
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tjpzq |
rjrvn tqiyl• kzvévn |
én¡ntqn vzsq·tvn |
vz I (Q) = UQ− |
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Ìrjo€kjnzx¹ zjrqu én¡ntqnu ¹ks¹nzx¹ I |
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(Q) = Q UQ− Æ xjuvu |
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lnsn nxsq rkjléjztvn yéjktntqn U = D U + D qunnz lkj xvkwj
ljí•q} rvét¹ U |
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= U zv wv znvénun Æqnzj D |
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= U D |
= − U |
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Ívëzvuy tj¡n yéjktntqn ojwqx€kjnzx¹ |
zjr |
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U = |
U U − U ¬ |
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zvmlj énryééntztvn xvvztv¡ntqn qunnz zjrvp kql |
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I(Q + ) = U |
I(Q + ) − U I(Q) |
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Íévknéqu ·zv I |
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(Q) = Q UQ− |
lnpxzkqzns•tv ¹ks¹nzx¹ nmv én¡n |
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tqnu ΀ qunnu |
I (Q + ) = (Q + )UQ+ |
j I |
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(Q + ) = (Q + )UQ |
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Ívlxzjks¹¹ ëzq otj·ntq¹ k xvvztv¡ntqn wvsy·jnu v·nkql |
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tvn zv lnxzkv (Q + |
)UQ+ = (Q + )UQ+ − QUQ+ dtj·qz |
QUQ− |
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² kzvévn én¡ntqn xvvztv¡ntq¹ |
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dtj¹ lkj én¡ntq¹ I |
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(Q) = UQ− |
q I (Q) = QUQ− xvvztv¡ntq¹ |
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u€ y n uv nu tjwqxjz• nmv vi•nn én¡ntqn |
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I(Q) = & |
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UQ− |
+ & |
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QUQ− |
= (& |
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+ & |
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Q) UQ− |
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Ævz znwné• wyznu wvlivéj & q & uv tv ylvksnzkvéqz• síi€u tj·js•t€u yxsvkq¹u
fsy·jp éjkt€} rvétnp }jéjrznéqxzq·nxrvmv yéjktntq¹ |
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Éqtnpt€n énryééntzt€n xvvztv¡ntq¹ x wvxzv¹tt€uq rvë{ {q|qntzjuq wvé¹lvr rvzvé€} ivs•¡n lky} én¡jízx¹ zjrqu
n xwvxvivu Íyxz• xvvztv¡ntqn qunnz kql
I(Q + N) = D I(Q + N − ) + + DN I(Q)
fvxzjks¹nu }jéjrznéqxzq·nxrvn yéjktntqn UN = D UN− + + D |
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cxsq kxn rvétq U |
U |
ëzvmv jsmniéjq·nxrvmv yéjktntq¹ N Np |
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N |
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xznwntq éjosq·t€ zv vi•nn én¡ntqn xvvztv¡ntq¹ qunnz |
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kql I(Q) |
= & |
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UQ− + & |
UQ− + + & UQ− |
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N N |
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cxsq |
n tjwéquné U = U = = UV zv ëzvuy rvétí xvvz |
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knzxzkyíz én¡ntq¹ |
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I (Q) = UQ− I |
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(Q) = Q UQ− I |
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(Q) = Q UQ− |
I |
(Q) = QV− UQ− |
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V |
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énryééntztvmv xvvztv¡ntq¹ Æ vi•nu én¡ntqq ëzvuy rvétí xvvzknzxzkynz ·jxz• (& + & Q + & Q + + &V QV− )UQ−
fvxzjks¹¹ zjrqn k€éj ntq¹ ls¹ kxn} rvétnp q xrsjl€kj¹ q} wvsy·jnu vi•nn én¡ntqn xvvztv¡ntq¹
Ëjwéquné én¡qu énryééntztvn xvvztv¡ntqn
I(Q + ) = I(Q + ) − I(Q + ) − I(Q + ) + I(Q)
®jéjrznéqxzq·nxrvn yéjktntqn qunnz olnx• kql U − U + U + U − =
èn¡j¹ nmv wvsy·jnu rvétq U = U = U = U = − dtj
·qz vi•nn én¡ntqn tj¡nmv xvvztv¡ntq¹ uv tv ojwqxjz• k xsn lyí•nu kqln I(Q) = (& + & Q + & Q ) Q + & (− )Q
djlj·q
Ëjplqzn vi•nn én¡ntqn énryééntzt€} xvvztv¡ntqp
j DQ+ − DQ+ + DQ = n DQ+ − DQ+ + DQ+ − DQ = i DQ+ + DQ+ − DQ =
k DQ+ − DQ+ + DQ =
m DQ+ + DQ =
l DQ+ + DQ+ + DQ =
Ëjplqzn DQ otj¹ énryééntztvn xvvztv¡ntqn q tj·js•t€n
·snt€
j DQ+ − DQ+ + DQ = D = D = − i DQ+ − DQ+ + DQ = D = D =
-sjkj 9,, иcЗПииcЛТЛФc fММТЛМ‘cЛК‰
k DQ+ + DQ+ + DQ = D = − D = −
m DQ+ − DQ+ + DQ+ − DQ = D = D = − D = −
Ëjplqzn zjryí wvxsnlvkjzns•tvxz• ·zv D = FRV ϕ D = FRV ϕ
q DQ+ − DQ+ FRV ϕ + DQ =
bvrj qzn ·zv wvxsnlvkjzns•tvxz• x vi•qu ·sntvu DQ = QW wéq
≤ W ≤ N − ylvksnzkvé¹nz xvvztv¡ntqí |
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D |
Q+N |
− & D |
+ & D |
Q+N− |
− + (− )N&ND |
Q |
= |
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N Q+N− |
N |
N |
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