Веленкин;Комбинаторика
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síi€u viéjovu éjxwvsv qz• zéq |knzj vzsq·t€n vz |knzj ymsvkvmv rkjléjzj = xwvxvivk Íénlwvsv qu ·zv k€iéjt€ |knzj yrj
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Òjr rjr k knézqrjs•t€} q mvéqovtzjs•t€} é¹lj} lvs t€ i€z• wénlxzjksnt€ kxn |knzj zv kv kzvévu é¹ly uv nz i€z• vltj qo xsn lyí•q} rvuiqtj|qp |knzvk ïÈfÇ ïÇfÈ ïfÈÇ Æ wnékvu qo ëzq} kjéqjtzvk vltvotj·tv vwénlns¹nzx¹ vréjxrj rsnzvr kzvévmv knézqrjs• tvmv é¹lj q vxzjízx¹ lkn kvouv tvxzq vréjxrq vxzjk¡q}x¹ rsnzvr
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l€p qo vxzjs•t€} lky} kjéqjtzvk wéqkvlqz sq¡• r vltvp kvo |
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tvxzq vréjxrq Æxnmv ( + ) = xwvxvivk vréjxrq |
Çj lj¹ uvtnzj uv nz wvwjxz• k vlqt qo lky} rjéujtvk Ívë zvuy qunnu xwvxvivk = xwvxvivk
Òjr rjr rj l€p xzylntz uv nz wvsy·qz• vlty qo zén} v|ntvr zv qunnu = xwvxvi xlj·q ërojuntvk
Æ zv·rj} wyzq nxz• k€ivé un ly lkyu¹ kvouv tvxz¹uq Ívëzvuy ·qxsv wyznp éjktv =
Òjr rjr k€ivé k tj·js•tvp zv·rn y n xlnsjt zv vxzjnzx¹
= kvouv tvxznp =
Ëvunévk x vltvp iyrkvp x lkyu¹ q x zénu¹
Æxnmv wv wéjkqsy xyuu€ tvunévk
j = tvunévk i =
dj¡q{éynu rj l€p tjivé oyivk wvxsnlvkjzns•tvxz•í tysnp q nlqtq| tys• xzjkqzx¹ nxsq tj ljttvu unxzn tnz oyij q nlqtq|j
nxsq nxz• ïqxsv zjrq} wvxsnlvkjzns•tvxznp éjktv Òjr rjr rj |
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lvuy qznsí xvvzknzxzkynz xkv¹ wvxsnlvkjzns•tvxz• zv ·qxsv |
qznsnp |
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tn ivs•¡n ·nu = |
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Òjr rjr x wvuv••í lky} |q{é q |
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tv tjwqxjz• N |
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N otj·t€} ·qxns zv vi•nn rvsq·nxzkv |
qxrvu€} |
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éjktv |
+ + + + + = bs¹ zén} |q{é jtjsvmq·tv wvsy·jnu+ + + + + = cxsq ls¹ wnékvp |q{é€ nxz• sq¡•kvouv tvxzq zv wvsy·jnu ( + + + + + ) = ·qxns
Çj l€p ítv¡j uv nz k€iqéjz• qo unxz éjivz€ j rj lj¹ lnky¡rj ² qo unxz Æxnmv wvsy·jnu = xwvxvivk k€ivéj
Çj lj¹ ryéq|j uv nz sqiv kvpzq sqiv tn kvpzq k ·qxsv
k€iéjtt€} Ívëzvuy qunnu xwvxvivk k€ivéj ryé Òjr rjr wv yx svkqí }vz¹ i€ vltj ryéq|j lvs tj i€z• k€iéjtj wvsy·jnu xwv
xvivk k€ivéj ryé Òv·tv zjr n nxz• − = xwvxvivk k€ivéj yzvr q − = xwvxvij k€ivéj myxnp Æxnmv = xwvxvivk
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Æ wnékvu xsy·jn P xwvxvivk kv kzvévu ( )P = P znwné• ls¹ |
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rj lvmv__qo P |knzvk kxn} xknzv{vévk nxz• k€ivé mvénz• |
tn mvénz• |
$ − $ Æ éyxxrvu js{jkqzn iyrk€ tv wv réjptnp unén x iyrk Ó h quntj tn tj·qtjízx¹ Ívëzvuy vi•nn ·qxsv éjo
sq·t€} qtq|qjsvk tn ivs•¡n = ·zv unt•¡n
Ív wéjkqsy wévqoknlntq¹
rtqm uv tv wnénwsnxzq k wnénwsnz€ |knzvk xwvxvijuq Êo tq} tn ivsnn ·nu k lkj |knzj rtqmq iylyz wnénwsnznt€ k
xsy·j¹} xwvxvijuq k€iqéjnzx¹ vzxyzxzkyí•qp |knz éjo ² vlqt qo qxwvs•oynu€} |knzvk j k xsy·j¹} ² k vlqt |knz Ív {véuysn
krsí·ntqp q qxrsí·ntqp k − + = xsy·j¹} rtqmq iylyz wnénwsnznt€ k wnénwsnz€ kxn} zén} |knzvk
Ív {véuysn krsí·ntqp q qxrsí·ntqp ·qxsv éjivzjí•q} éjktv + + − − − + = Òvs•rv jtmsqpxrqp ¹o€r otjíz
− − + = zvs•rv {éjt|yoxrqp otjíz − − + =
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Çvsq·nxzkv ·qxns vz lv lns¹•q}x¹ tj éjktv > @ |
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mln >[@ ² |nsj¹ ·jxz• |
[ Òv·tv zjr n tj lnsqzx¹ > @ tj |
² > @ ·qxns Ív {véuysn krsí·ntqp q qxrsí·ntqp wvsy·jnu |
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·zv − > @ − > @ + > @ = ·qxns tn lnsqzx¹ tq tj |
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tq tj Òv·tv zjr |
n tj}vlqu ·zv ·qxns tn lnsqzx¹ tq tj |
tq tj tq tj tq tj |
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Ëj wnékvu unxzn uv nz xzv¹z• síij¹ qo |q{é
tj kzvévu ² síij¹ qo vxzjk¡q}x¹ |q{é tj zénz•nu ² síij¹ qo|q{é q z l Æxnmv wvsy·jnu = ·qxns
bvwvstqu ·qxsj vz lv kwnénlq tys¹uq lv xnuq otj·t€} Çvsq·nxzkv zjrq} xnuqotj·t€} ·qxns ojwqx• rvzvé€} tn
xvlné qz nlqtq|€ éjktv dtj·qz un ly q qunnzx¹− = ·qxns ojwqx• rvzvé€} tn xvlné qz êzv unt•¡n
·nu wvsvkqtj
Çj lj¹ qo P wé¹u€} wévknlntt€} ·néno tnrvzvéyí zv·ry $ wnénxnrjnzx¹ x P wé¹u€uq Ívëzvuy wé¹u€n wév}vl¹•qn ·néno
zv·ry $ ljíz P zv·nr wnénxn·ntq¹ Ìi•nn ·qxsv zv·nr wnénxn·ntq¹ vzsq·t€} vz ljtt€} zén} éjktv P
Íévkvlqu tnwvxénlxzkntt€p wvlx·nz x y·nzvu zén} uvuntzvk wv¹ksntq¹ k ·qxsn tvkvp |q{é€ êzvz uvuntz iylnz tj p |q{én kjéqjtzvk wnékvp |q{é€ ² síij¹ révun tj p kjéqjtzvk ² síij¹ révun wnékvp q n•n tj vltvu unxzn kvxnu• n•n tn qxwvs• ovkjtt€} |q{é léymqn |q{é€ wéqlnzx¹ k€iqéjz• qo ·qxsj y n wv ¹kqk¡q}x¹ êzv wéqkvlqz r vzknzy k kqln xyuu€ xsjmjnu€}
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·nsvknr ènjsqoynzx¹ uqtqujs•t€p kjéqjtz ² nxsq x léy mq} ryéxvk i€sv }vz¹ i€ wv vltvuy xzylntzy zv kunxzn vtq wéqlyujsq+ + + = ojlj· + =
Ëj u unxzn uv nz xzv¹z• vltj qo |q{é tj u u u q u ² vltj qo |q{é j tj wvxsnltnu ² vltj qo |q{é nn
·nztvxz• vwénlnsntj Æxnmv wvsy·jnu = ·qxns cxsq
ko¹z• kxn ·qxsj vz lv wvsy·qu ·qxns
Çvsq·nxzkv w¹zqotj·t€} ·qxns éjktv qo tq} rj lj¹ |q{éj ¹ks¹nzx¹ ·nzt€u ·qxsvu k = xsy·j¹} tn·nzt€u ²
k = xsy·j¹} gq{é€ unt•¡qn ·nu tn k}vl¹z k =
xsy·j¹} j ivs•¡qn ·nu ² k = xsy·j¹}
bvwvstqu kxn ·qxsj kwnénlq tys¹uq lv otj·t€} q n•n krsí
·qu ·qxsv ïqxns iylnz ls¹ q} ojwqxq |
wvzéniynzx¹ |
|q{é j wvxrvs•ry kxn |q{é éjktvwéjkt€ q lvs |
t€ k}vlqz• |
vlqtjrvkvn ·qxsv éjo rj lj¹ iylnz ywvzénisntj éjo k vi•nu xsy·jn Q Q− Ëysnp k N otj·tvu ·qxsn iylnz (N − ) N− wnékj¹
|q{éj ² síij¹ qo lnk¹zq k rj lvu kjéqjtzn oj tní wévqokvs•tvn (N − ) otj·tvn ·qxsv k tn ivsnn ·nu otj·tvu ²
+ + + + =
fyuuj kxn} ·nz€én}otj·t€} ·qxns vz lv éjktj
+ = lns¹•q}x¹ tj ² ylkvnttvp xyuun ·qxns vz = lv = q} − + = z n
+
xyuuj = ¬tjsvmq·tv xyuuj ·qxns
lns¹•q}x¹ tj éjktj tj ² tj ²
tj ² tj ² tj ² wv {véuysn krsí·ntqp q qxrsí·ntqp wvsy·jnu vzknz
ftj·jsj k€inénu wvxsnltíí |q{éy éjxxuvzénk kjéqjtzj q tn ojznu wnékyí ní tn uv nz i€z• ojznu vxzjs•t€n Ìzknz wvsy·qzx¹ k kqln +
èvktv wvsvkqtj qo kxn} Q xsvk z n Q− bs¹ |
Q tn·nztvmv |
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ëzv v·nkqltv ojuntvp tj j tj qo xsvkj x ·nzt€u ·qxsvu |
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tysnp q tn·nzt€u ·qxsvu nlqtq| wvsy·jnzx¹ xsvkv x tn·nzt€u ·qx |
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svu tysnp wvëzvuy xsvk x ·nzt€u ·qxsvu tysnp xzvs•rv |
n xrvs•rv |
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x tn·nzt€u bs¹ Q ·nztvmv rj |
lvn xsvkv qunnz kql q ojznu xsvkv |
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lsqt€ Q − x tn·nzt€u ·qxsv |
tysnp qsq q ojznu |
xsvkv lsqt€ |
Q − x ·nzt€u ·qxsvu tysnp k rj lvu kjéqjtzn Q− kvouv tvxznp |
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qzvmv Q− = Q− |
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djwqx• tj xzvsin qunnz kql |
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y n lkn éjosq·t€n |q{é€ |
Íéq |
ëzvu nxsq vltv |
qo |
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untnn ·nu zén}otj·tv zv xénlq |q{é nxz• j zvmlj q q uv tv lvwvstqz• ojwqx• kwnénlq tys¹uq lv zén}otj·tvp dtj·qz xwvxvij
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uq uv tv k€iéjz• wjéy |q{é q q q q q q
xwvxvijuq ² wnékvn ·qxsv tj rj lvn qo zén} unxz vltj qo lky} |q{é kzvévn ·qxsv vwénlns¹nzx¹ wnék€u qzvmv = xwvxvivk
cxsq wnék€p ·snt wévménxxqq éjknt D j nn éjotvxz• G zv zénzqp ·snt éjknt D + G Ív yxsvkqí D + G ≤ Q êzv tnéjkntxzkv
wéq ojljttvu G ≤ G ≤ Q − qunnz Q − G én¡ntqp Æxnmv wvsy·jnu ( Q − ) + ( Q − ) + + = Q(Q − ) én¡ntqp bs¹ wvxsnlvkjzns•tvxzq
·qxns Q + qunnu Q wévménxxqp
bs¹ vwénlnsnttvxzq iylnu éjxwvsjmjz• ·qxsj E E E zjr ·zvi€ vtq viéjovk€kjsq kvoéjxzjí•yí mnvunzéq·nxryí wévménxxqí
Íyxz• T = E = Q > P q Q kojqutv wévxz€ Òvmlj wévménxxqí uv tv
E P
ojwqxjz• k kqln EP EPQ EQ wéq·nu E P Q ² tjzyéjs•t€n ·qxsj
EQ ≤ P < Q bs¹ rj lvmv Q vz |
lv qo tnéjkntxzkj E ≤ |
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tj}vlqu kvouv |
t€n otj·ntq¹ E qo |
yxsvkqp P < Q P q Q kojqutv |
wévxz€ ² kvouv |
t€n otj·ntq¹ P tjwéquné wéq Q = iylnz E vz |
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lv q P = qsq P = wnénutv jnu q xrsjl€kjnu ls¹ kxn} Q
Ívsy·jnu |
+ + + + + + + + = |
kjéqjtzvk |
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tv x·qzjz• ·zv kné}tnu mvéqovtzjs•tvu é¹ly q k snkvu |
knézqrjs•tvu nxz• vxkn•nttvn vrtv cxsq kné}tnn snkvn ymsvkvn vrtv |
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vxkn•ntv zv ls¹ vxzjs•t€} vrvt nxz• PQ− kjéqjtzvk cxsq vtv tn |
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vxkn•ntv zv xknzqzx¹ }vz¹ i€ vltv qo Q − vxzjk¡q}x¹ kné}tq} vrvt |
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Q− − kjéqjtzvk q }vz¹ i€ vltv qo P − vxzjk¡q}x¹ snk€} vrvtP− − kjéqjtzvk bs¹ vxzjk¡q}x¹ vrvt nxz• (P− )(Q− ) kjéqjtzvk Êzvmv PQ− + ( P− − )( Q− − ) (P− )(Q− ) kjéqjtzvk
bs¹ rj lvp wjé€ wénlunzvk nxz• zéq kvouv tvxzq ² k k€ ivéry uvmyz kvpzq lkj vlqt qsq tq vltvmv wénlunzj qo ëzvp wjé€
Ívëzvuy ·qxsv k€ivévr éjktv Æ vi•nu xsy·jn Q
Òjr rjr ·qxsj tn uvmyz tj·qtjz•x¹ x tys¹ zv qunnu− = ·qxns Êsq wnékyí |q{éy uv tv k€iéjz• xwvxvijuq
kxn révun rj lyí qo xsnlyí•q} zén} ² xwvxvijuq qzvmv= xwvxvivk
Òjr rjr xv kxnuq ¡nxz•í léyo•¹uq ¹ vinljs vlqt éjo j x rj l€uq w¹z•í ² lkj éjoj zv n•n wv vltvuy éjoy ¹ vinljs x rj lvp w¹znérvp léyonp k vzxyzxzkqn ¡nxzvmv Ëv zvmlj tj ëzq} xnuq vinlj} x rj l€uq ·nz€é•u¹ ¹ vinljs qunttv éjoj lkj vinlj kw¹znévu q vlqt k¡nxznévu x rj l€uq zénu¹ ² éjoj q x rj l€uq lkyu¹ ²
éjo f rj l€u qo léyonp tj ëzq} vinlj} ¹ kxzénzqsx¹ éjo dtj·qz n•n wv vltvuy éjoy ¹ vinljs klkvnu x rj l€u léymvu Çj l€p léym vzxyzxzkvkjs tj qo ëzq} vinlvk vinlj} klkvnu x léymquq léy o•¹uq q vinln kw¹znévu Òjr rjr ino rj lvmv qo léyonp ¹ vinljs
éjo zv éjoj ¹ vinljs vlqt j kxnmv i€sv vinlvk
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ÌÒÆcÒÔ __ |
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cxsq ljttvn ·qxsv qunnz kql DE = D + E |
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qk nmv x |
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ED = E + D wvsy·qu (D + E) Òjr rjr |
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rkjléjz j ≤ D + E ≤ zv D + E = Ívsy·jnu kvouv |
tvxznp |
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Ëjqunt•¡nn ·qxsv qunnz kql k rvt|n ² ·zv wvsy·qz |
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x¹ tjqivs•¡nn ² |
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j Ëjwéquné i fyuuj kxn} lnk¹zq ojljtt€} |
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·qxns éjktj Æv kxn xyuu wv ·qxsj kvplnz ·qxns rj |
lvn |
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qo ojljtt€} wv |
éjoj xyuuj xyuu lvs tj éjkt¹z•x¹ |
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$ = xsvkjénp $ |
− $ = |
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Òjr rjr jwns•xqt€ éjosq·t€ zv qunnu $ = xwvxvivk
Æv kzvévu xsy·jn rj l€p jwns•xqt uv nz wvwjxz• síivuy qo x€tvknp wvëzvuy qunnu = xwvxvivk
ïj¡rq uvmyz i€z• éjxxzjksnt€ $ isíl|j $ q sv rq $
xwvxvijuq Æxnmv wv wéjkqsy wévqoknlntq¹ $ $ $ = xwvxvivk
Òéq xyl•q uvmyz k€iéjz• wvinlqzns¹ xwvxvijuq Òén} éjo
sq·t€} rjtlqljzvk vtq tjovkyz k $ = xsy·j¹} Ívëzvuy xvkwj
lntqn }vz¹ i€ y lky} xylnp iylnz k xsy·j¹} bvs¹ zjrq} xsy·jnk éjktj
Ívsy·jnu éjoun•ntq¹ x wvkzvéntq¹uq qo rjéz wv Æxnmv
= xwvxvi cxsq xénlq rjéz tn lvs tv i€z• wjé zv qunnu
éjoun•ntq¹ ino wvkzvéntqp q} ·qxsv $ =
Æ wnékvu xsy·jn lvxzjzv·tv k€iéjz• vlty wqrvkyí q vlty iyi tvkyí rjézy wvsy·jnu = xwvxvivk Æv kzvévu xsy·jn k€iqéjnu
wqry q zén{y q nxsq vtq éjotvp knsq·qt€ zv wjéy ls¹ wqrq iyit€ qsq ·nékq cxsq vtq vlqtjrvkvp knsq·qt€ zv ivs•¡n k€iqéjz• tn ·nmv Êzvmv + =
__ kjéqjtzvk
Æxnmv zjrq} ·qxns $ = k tq} |q{é rj lj¹ qo |q{é
ywvzénis¹nzx¹ éjoj ² k rj lvu qo éjoé¹lvk = éjo wvë zvuy xyuuj |q{é wnékvmv éjoé¹lj ljxz ( + + + ) = kzvévmv ² q zénz•nmv ² fyuuj éjktj
cxsq |q{é€ tn wvkzvé¹ízx¹ zv zjrq} ·qxns $ = k tq} |
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|q{é€ rj lj¹ qo |
|q{é ywvzénis¹nzx¹ k |
rj lvu qo éjoé¹lvk |
éjo wvëzvuy xyuuj |
( + + + )( + + |
) = |
ïqxsv lvs tv vrjt·qkjz•x¹ vlty qo w¹zq rvuiqtj|qp
wnék€n n lkn |q{é€ uvmyz i€z• wévqokvs•t€uq Æxnmv wvsy·jnu = ·qxns Æv kzvévu xsy·jn ·qxsv lvs tv vrjt·q
kjz•x¹ vlty qo ·nz€én} rvuiqtj|qp wnék€n n lkn
ÌÒÆcÒÔ
|knzj uvmyz wvkzvé¹z•x¹ zv vxy•nxzks¹¹ k€ivé xkné}y ktqo qunnu= xwvxvivk
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zv·nr wnénxn·ntq¹ & (& − ) |
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Êo w¹zq ltnp tjlv k€iéjz• lkj k rvzvé€n ljízx¹ ¹isvrq |
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Æxnmv & = xwvxvivk Æ vi•nu xsy·jn &N |
xwvxvivk |
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Q+N |
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= $ = |
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Ìi•nn ·qxsv xwvxvivk k€tyz• |
rjéz éjktv & ïqxsv |
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xwvxvivk wéq rvzvé€} tn k€iqéjíz tq vltvmv zyoj éjktv & Ívë
zvuy }vz¹ i€ vlqt zyo iylnz k & − & xsy·j¹} èvktv vlqt zyo k |
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évktv lkj zyoj k & |
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wv ·nz€énu ujxz¹u uvmyz |
éjxwénlnsqz•x¹ zjr |
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+ + + qsq + + + f y·nzvu k€ivéj vltvp qsq lky} ujxznp |
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(& ) + & (& ) (& ) |
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