Веленкин;Комбинаторика
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xsy·jn u€ qunsq i€ |
E + F = Q − |
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E ≠ F |
tv E + > F |
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D + E > F wéq·nu ·qxsj D + E q F quníz vlqtjrvkyí ·nztvxz• dtj·qz
D + E ≥ F + Ëv zvmlj ·qxsj D − E − F − viéjoyíz éjoiqntqn ls¹Q − wéq·nu (D − ) + (E − ) > F − Ònu xju€u yxzjtjksqkjnzx¹ koj
qutv vltvotj·tvn xvvzknzxzkqn un ly zénymvs•tqrjuq x wnéqunzéjuqQ q Q −
Ëju tjlv tjpzq rvsq·nxzkv zévnr tjzyéjs•t€} ·qxns [ \ ] zjrq} ·zv [ ≤ \ ≤ ] [ + \ + ] = q [ + \ > ] Êo ëzq} tnéjkntxzk k€
znrjnz ·zv ] uv nz wéqtqujz• otj·ntq¹ ylvksnzkvé¹í•qn tnéjknt xzkju ≤ ] ≤ cxsq ] = zv [ + \ = wéq·nu [ ≤ \ ≤ Ívë
zvuy ≤ \ ≤ q u€ qunnu zénymvs•tqrvk x ] = Òv·tv zjr n yxzjtjksqkjnu ·zv ·qxsv zénymvs•tqrvk ls¹ rvzvé€} ] =
éjktv xvvzknzxzknttv j kxnmv qunnu zénymvs• tqrj
Òv·tv zjr n wvsy·jnu ·zv ·qxsv zénymvs•tqrvk x wnéqunzévuéjktv
Ævo•unu zénymvs•tqr x wnéqunzévu Q Íyxz• nmv xzvévt€ éjkt€ [ \ ] Íéqijks¹¹ r lsqtju ëzq} xzvévt wv wvsy·qu ·qxsj [ + \ + ] + ¹ks¹í•qnx¹ lsqtjuq xzvévt zénymvs•tqrj x wnéq
unzévu Q + Ëv révun zvmv qunnu zénymvs•tqrq xv xzvévtjuq ( Q + Q + ) ( Q Q + ) (Q + Q + Q + ) rvzvé€n tn
uvmyz i€z• wvsy·nt€ vwqxjtt€u viéjovu wvxrvs•ry wéq viéjztvu wév|nxxn xyuuj lky} unt•¡q} xzvévt xzjtvkqzx¹ unt•¡n zénz•np xzv évt€
Íyxz• 1 = Q Ëju tjlv tjpzq rvsq·nxzkv zévnr tjzyéjs•t€} ·qxns [ \ ] zjrq} ·zv [ ≤ \ ≤ ] [ + \ + ] = Q q [ + \ > ] Êo ëzq} tnéjkntxzk xsnlynz ·zv Q ≤ ] ≤ Q − Íéq ëzvu nxsq ] = N zv [ + \ = Q − N j ·qxsv |nsv·qxsntt€} én¡ntqp ëzvmv yéjktntq¹ zj
rq} ·zv [ ≤ \ ≤ ] = N éjktv N − Q + cxsq |
n ] = N + zv qunnu |
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N − Q + én¡ntqp Ívëzvuy ·qxsv zénymvs•tqrvk éjktv |
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Q− |
Q− |
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∑ ( N − Q + ) + ∑ ( N − Q + ) = Q |
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N= Q |
N= Q |
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Ìxzjs•t€n xsy·jq éjoiqéjízx¹ zv·tv zjr |
n Íéq wnén}vln vz |
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1 r 1 + uv tv qxwvs•ovkjz• én¡ntq¹ wénl€ly•q} ojlj· |
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Ìméjtq·ntq¹ |
≤ D ≤ E ≤ O D + E ≥ O + |
ojljíz tj wsvxrvxzq |
2DE ymvs•tqr qo |nsv·qxsntt€} zv·nr (D E) ·qxsv zv·nr k rvzvévu
snmrv wvlx·qz€kjnzx¹
Êxwvs•oypzn wénl€ly•yí ojlj·y
Íé¹u€n wnékvmv wy·rj éjoiqkjíz wsvxrvxz• tj P ·jxznp Ínékj¹ wé¹uj¹ kzvévmv wy·rj wnénxnrjnzx¹ xv kxnuq P wé¹u€uq wné kvmv wy·rj q ljnz P + tvk€} ·jxznp Æxn vxzjs•t€n wé¹u€n kzvévmv
wy·rj quníz P + zv·nr wnénxn·ntq¹ x éjtnn wévknlntt€uq wé¹u€uq Ívëzvuy u€ qunnu kxnmv P + P + + (Q − )(P + ) ·jxznp
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bvrj nu yzkné lntqn x wvuv••í qtlyr|qq wv ·qxsy V réq k€} cxsq V = zv vtv v·nkqltv zjr rjr zv·nr wnénxn·ntq¹ tnz q
·qxsv visjxznp éjktv
Íyxz• yzkné lntqn y n lvrjojtv ls¹ V réqk€} Ævo•unu xqxznuy qo V réqk€} quní•q} QU zv·nr wnénxn·ntqp réjztvxzq U kxíly
U = Òvmlj vtq vméjtq·qkjíz + Q + Q + + UQU+ + oju rtyz€} visjxznp Íévknlnu réqkyí V + wyxz• vtj qunnz NU zv·nr
wnénxn·ntq¹ réjztvxzq U x éjtnn wévknlntt€uq réqk€uq j kxnmv N + N + + NU+ + zv·nr wnénxn·ntq¹ êzq zv·rq wnénxn·ntq¹ éjo
iqkjíz nn tj N + N + + NU+ + ·jxznp Çj lj¹ ·jxz• wévknlnttvp
réqkvp xvvzknzxzkynz vltvp tvkvp visjxzq j wvzvuy znwné• ·qxsv vi sjxznp vrj nzx¹ éjkt€u
( + Q + Q + + UQU+ + ) + (N + N + + NU+ + ) |
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Ëv nxsq tvkj¹ réqkj¹ wév¡sj ·néno zv·ry wnénxn·ntq¹ réjztvxzq
U éjtnn wévknlntt€} réqk€} zv znwné• ëzj zv·rj xzjsj zv·rvp wnén xn·ntq¹ réjztvxzq U +
Ìivotj·qu ·néno QU′ rvsq·nxzkv zv·nr wnénxn·ntq¹ réjztvxzq U k
tvkvp xqxznun réqk€} ‰xtv ·zv QU′+ = QU+ − NU+ + NU+ vz éjtnn qunk¡q}x¹ zv·nr wnénxn·ntq¹ réjztvxzq U + tjlv vzt¹z• NU+ zv·nr qunk¡q} réjztvxz• U + yknsq·qk¡q} nn q wéqijkqz• NU+ zv·nr qunk¡q} réjztvxz• U q yknsq·qk¡q} nn Ëv zvmlj
+ Q′ + Q ′ + + UQU′+ + =
= + (Q − N + N ) + (Q − N + N ) + U(QU+ − NU+ + NU+ ) + =
= ( + Q + Q + + UQU+ + ) + (N + N + + NU+ + )
΀ lvrjojsq zjrqu viéjovu ·zv + Q′ + Q ′ + + UQU′+ + éjktv
·qxsy visjxznp ls¹ tvkvp xqxznu€ réqk€} xu {véuysy Æ xqsy wéqt|qwj ujznujzq·nxrvp qtlyr|qq yzkné lntqn knétv ls¹ síivmv ·qxsj réqk€}
Íyxz• Q ·nztv Q = N Òvmlj Q xsnlyí•quq xwvxvijuq uv nz i€z• wénlxzjksntv k kqln xyuu€ lky} xsjmjnu€}
Q = + ( N − ) = + ( N − ) = = N + N
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k€iéjz• &N xwvxvijuq Ívëzvuy kxnmv qunnu
ÌÒÆcÒÔ
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(N − ) |
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( N − ) + ( N − ) + + (N − )(N + ) + |
N |
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N− |
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Q(Q − ) |
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= ∑ V( N − V) + |
N(N − ) |
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N(N − ) |
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V= |
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xwvxvivk wvsy·qz• xyuuy Q = N cxsq |
n Q tn·nztv |
Q = N − zv |
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Q = + ( N − ) = + ( N − ) = = (N − ) + N q ·qxsv xwvxvivk éjktv
N− |
N(N − )( N − ) |
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Q(Q − ) |
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∑ V( N − V − ) = |
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j Íyxz• tj wsvxrvxzq wévknlntj N − wé¹uj¹ Íévknlnu n•n
vlty wé¹uyí Ìtj éjoiqkjnzx¹ zv·rjuq wnénxn·ntq¹ x éjtnn wévknlnt |
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t€uq wé¹u€uq tj N |
·jxznp rj lj¹ qo rvzvé€} xvvzknzxzkynz vltvuy |
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tvkvuy ryxry wsvxrvxzq wéq ëzvu wé¹u€} éjoiqkjíz wsvxrvxz• tj |
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·jxz• ÊFwvs•oy¹ |
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Q wé¹u€} éjoiqkjíz wsvxrvxz• tj + ( + + + Q) = |
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i Ìivotj·qu ·néno 3(Q) ·qxsv ·jxznp tj rvzvévn Q wsvxrvxznp lns¹z wévxzéjtxzkv Íyxz• y n wévknlntj N − wsvxrvxz• Íévknlnu n•n vlty N í wsvxrvxz• Ìtj wnénxnrjnzx¹ x éjtnn wévknlntt€uq wsvx
− N +
rvxz¹uq wv N − wé¹uvp rvzvé€n lns¹z nn tj N ·jxznp Çj lj¹ qo ëzq} ·jxznp xvvzknzxzkynz tvkvp ·jxzq wévxzéjtxzkj Ívëzvuy
qunnu 3(N) = 3(N − ) + N − N + wéq·nu wsvxrvxznp lns¹z wévxzéjt
Q
xzkv tj ·jxz• wvëzvuy 3(Q) = + ∑ (N − N + ) ·jxznp Íéqunt¹¹N=
{véuys€ xyuu€ ·qxns vz lv Q q xyuu€ rkjléjzvk ·qxns vz lv
Q xu w tj}vlqu ·zv 3(Q) = (Q + )(Q − Q + )
cxsq tn wéqunt¹z• ëzq {véuys€ tv otjz• ·zv xyuuj V } xznwntnp Q iylnz utvmv·sntvu (V + ) p xznwntq vz Q zv unzvlvu tnvwénlnsnt
t€} rvë{{q|qntzvk wvsy·qu ·zv 3(Q) = $Q + %Q + &Q + ' wéq·nu 3( ) = 3( ) = 3( ) = 3( ) = vzrylj tj}vl¹zx¹ $ % & q '
Íéq lvijksntqq (N + ) p wsvxrvxzq r éjtnn wévknlntt€u N wsvxrvxz¹u kvotqrjnz N tvk€} ·jxznp j + + + + + (Q − ) = = Q − Q +
Çj lj¹ ktyzéntt¹¹ zv·rj wnénxn·ntq¹ lqjmvtjsnp vltvotj·tv vwénlns¹nzx¹ ojljtqnu kné¡qt Q ymvs•tqrj ² rvt|vk wnénxnrjí
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•q}x¹ lqjmvtjsnp Ívëzvuy ·qxsv ktyzénttq} zv·nr wnénxn·ntq¹ lqj |
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mvtjsnp éjktv & = Q(Q − )(Q − )(Q − ) |
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Ëjplnu znwné• vi•nn ·qxsv zv·nr wnénxn·ntq¹ lqjmvtjsnp |
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rj lvp kné¡qt€ Q ymvs•tqrj k€}vl¹z Q − lqjmvtjsnp j kxnmv u€ |
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qunnu |
Q(Q − ) lqjmvtjsnp |
Çj lj¹ lqjmvtjs• $% wnénxnrjnzx¹ |
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kxnuq lqjmvtjs¹uq xvnlqt¹í•quq kné¡qt€ vzsq·t€n vz $ q |
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iylnz |
Q(Q − ) |
− (Q − ) + |
Ï·qz€kj¹ ·zv vi•nn ·qxsv lqjmvtjsnp |
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Q(Q − ) rj lj¹ zv·rj wnénxn·ntq¹ x·qzjnzx¹ wéq ëzvu lkj |
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zv·nr wnénxn·ntq¹ lqjmvtjsnp ² ktyzénttqn wvsy·jnu ·zv ·qxsv |
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(Q − )(Q − )(Q − ) |
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|qntzvk tjpzq ëzvz utvmv·snt 1(Q) = Q − Q + Q − Q +
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sy·jnu ° Òjr rjr xyuuj ymsvk rj lvmv zénymvs•tqrj éjktj
° zv wvsy·qzx¹ zénymvs•tqrvk Æ vi•nu xsy·jn Q + P − zénymvs•tqrj
cxsq lvijkqz• r kné¡qtju síivmv utvmvymvs•tqrj tn xvlné j•nmv zv·ry $ ëzy zv·ry wvsy·qu utvmvymvs•tqr xvlné j•qp
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( N + P)(P + ) = =
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tv ko¹z• qsq qmévrj F qsq qmévrj |
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