Веленкин;Комбинаторика
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fyuuqéy¹ énoys•zjz€ wvsy·ntt€n k lky} wvxsnltq} ojlj·j} wéq }vlqu r wvxsnltnp {véuysn
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cxsq i€ tqrjrqn zéq qo Q zv·nr tn sn jsq tj vltvp wé¹uvp zv i€sv i€ &Q zénymvs•tqrvk x kné¡qtjuq k ëzq} zv·rj} Ëv S zv·nr
sn jz tj vltvp wé¹uvp q wvëzvuy &S zénymvs•tqrvk tjlv vziévxqz•
Ìxzjnzx¹ & − & zénymvs•tqrvk
Q S
j Îv tv ko¹z• lkn kné¡qt€ tj vltvp wé¹uvp j zénz•í ²
tj léymvp Ívëzvuy iylnz & & + & & = ST (S + T − ) zénymvs•tqrvk |
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S T |
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c•n |
& (& + & ) + |
& (& + & ) |
+ & & & = |
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(S + T)(S + T + U − ) |
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zénymvs•tqrvk |
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Ëj |
rj |
lvp xzvévtn |
Q − zv·rj |
lnsntq¹ wéq·nu kné¡qt€ |
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zénymvs•tqrj tns•o¹ |
iéjz• |
kxn |
tj vltvp |
xzvévtn Ìzxílj wvsy·jnu |
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− & |
= (Q − ) ( Q − ) |
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vzknz & (Q− ) |
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Q− |
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Ævvi•n Q wé¹u€} k vi•nu wvsv |
ntqq quníz & zv·nr wnén |
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Q |
xn·ntq¹ Ëv |
S wé¹u€} wév}vl¹•q} ·néno zv·ry |
$ ljíz vlty zv·ry |
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wnénxn·ntq¹ kunxzv &S D T wé¹u€} wév}vl¹•q} ·néno zv·ry % ²
vlty zv·ry kunxzv & Ívëzvuy vxzjnzx¹ & − & − & + zv·nr wnén |
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xn·ntq¹ |
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Q S |
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ïqxsv zv·nr wnénxn·ntq¹ sn j•q} tj rj lvp qo wé¹u€} |
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wév}vl¹•q} |
·néno $ |
éjktv Q + S |
·néno % ² |
P + S |
q ·néno & ² |
P + Q Òjr |
rjr ·néno |
$ wév}vl¹z |
P wé¹u€} |
·néno |
% wév}vl¹z Q |
wé¹u€} q ·néno & ² S wé¹u€} zv vi•nn ·qxsv zv·nr wnénxn·ntq¹
éjktv |
>P(Q + S) + Q(P + S) + S(P + Q)@ = PQ + PS + QS |
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Êo tq} uv tv &PQ+PS+QS xwvxvijuq k€iéjz• zévpry zv·nr tv k |
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+ Q& |
+ S& |
P&Q+S |
P+S |
P+Q xsy·j¹} u€ ojknlvuv wvsy·qu zv·rq sn j•qn |
tj vltvp qo wévknlntt€} wé¹u€} Ívëzvuy ·qxsv zénymvs•tqrvk éjktv
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− P& |
− Q& |
− S& |
&PQ+PS+QS |
Q+S |
P+S |
P+Q |
Çj l€u zénu wé¹u€u xvvzknzxzkyíz ·nz€én rjxjí•qnx¹ q} vréy tvxzq Ívëzvuy kxnmv qunnu & vréy tvxznp
Æ€inénu lkn zv·rq tj wé¹uvp D q lkn zv·rq tj wé¹uvp E Êu xvvzknzxzkyíz lkn zv·rq wnénxn·ntq¹ wé¹u€} wév}vl¹•q} ·néno ëzq zv·rq zv·rj wnénxn·ntq¹ lqjmvtjsnp zéjwn|qq q zv·rj wnénxn·n
tq¹ ivrvk€} xzvévt Òjr rjr tj wnékvp wé¹uvp uv tv k€iéjz• &S wjé zv·nr j tj kzvévp &T zv ·qxsv zv·nr wnénxn·ntq¹ éjktv &S&T qo tq} &S&T un ly ojljtt€uq wé¹u€uq
ïqxsv wévknlntt€} vréy tvxznp éjktv &Q Ævo•unu vlty qo ëzq} vréy tvxznp wévknlnttyí ·néno zv·rq $ % & ΀ qunnu
-sjkj ,,
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&Q− vréy tvxznp tn wév}vl¹•q} tq ·néno vlty qo ëzq} zv·nr Æ€ |
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iéjttj¹ vréy tvxz• |
wnénxnrjnzx¹ x rj lvp qo |
tq} k lky} zv·rj} |
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bjsnn qunnu & &Q− |
vréy tvxznp wév}vl¹•q} |
·néno vlty qo zv·nr |
$ % & q tn wév}vl¹•q} ·néno lkn qo ëzq} zv·nr Ìtq ljíz wv vltvp zv·rn wnénxn·ntq¹ vzsq·tvp vz $ % & Ìxzjs•t€n vréy tvxzq wnén xnrjízx¹ x k€iéjttvp k lky} qo zv·nr $ % & Òjrqu viéjovu ljttj¹ vréy tvxz• ljnz
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+ & & |
= |
(Q − )(Q − )( Q − ) |
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&Q− |
Q− |
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zv·nr wnénxn·ntq¹ vzsq·t€} vz $ % & Æxnmv wvsy·jnu
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& |
(Q − )(Q − )( Q − ) |
= |
( Q − ) |
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zv·nr wnénxn·ntq¹ k vi•nu xsy·jn éjosq·t€} vzsq·t€} vz ojljt
t€} bvijks¹¹ ëzq Q |
zv·nr wvsy·jnu ·zv ·qxsv zv·nr wnénxn·ntq¹ |
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tn wénkvx}vlqz ( Q − ) & + Q |
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Q |
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Q zv·nr vwénlns¹íz &Q vréy |
tvxznp Êo q} ·qxsj &Q− wév |
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}vl¹z ·néno ljttyí zv·ry q &Q− ² ·néno ljtt€n lkn zv·rq Ívëzvuy |
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wé¹uj¹ |
wév}vl¹•j¹ |
·néno lkn ljtt€n |
zv·rq |
qunnz |
tn |
ivsnn |
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+ ( & |
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− & |
) + zv·nr wnénxn·ntq¹ x vréy |
tvxz¹uq Òjr rjr |
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&Q− |
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Q− |
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Q− |
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·néno |
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zv·nr |
wév}vlqz |
& |
wé¹u€} |
zv qunnu tn |
ivsnn |
·nu |
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& ( & |
+ & |
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+ ) |
zv·nr wnénxn·ntq¹ |
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Q− |
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Çj |
lj¹ wé¹uj¹ wnénxn·ntq¹ vwénlns¹nzx¹ lkyu¹ wsvxrvxz¹ |
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uq j rj |
lj¹ wsvxrvxz• ² zénu¹ ojljtt€uq zv·rjuq Íé¹u€n éjx |
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wjljízx¹ tj rsjxx€ k xvvzknzxzkqq x znu xrvs•rv zv·nr qo ojljí•q}
wnékyí wsvxrvxz• k}vlqz k ·qxsv zv·nr ojljí•q} kzvéyí wsvxrvxz•
Æxn zv·rq éjosq·t€ k &Q&Q− xsy·j¹} k€iqéjnu zéq zv·rq qo Q q
n•n zéq zv·rq qo vxzjk¡q}x¹ Q − wéq·nu wvé¹lvr k€ivéj tn qméjnz
évsq ² xzvs•rv |
n iylnz vi€·tv éjosq·t€} wé¹u€} wnénxn·ntq¹ |
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wsvxrvxznp |
cxsq |
vltj zv·rj |
k}vlqz k |
vin zévprq zv wvsy·jnu |
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&Q &Q− &Q− |
xwvxvivk k€ivéj |
q xzvs•rv |
n wé¹u€} wnénxn·ntq¹ j |
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nxsq lkn zv·rq k}vl¹z k vin zévprq zv vtq q vwénlns¹z wé¹uyí
wnénxn·ntq¹ tnojkqxquv vz vxzjs•t€} zv·nr ² wvsy·jnu & |
wé¹u€} |
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Æxnmv qunnu (& & |
+ & & |
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) + & wé¹u€} |
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Q Q− |
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Q |
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& − & |
+ Çj |
lj¹ |
wsvxrvxz• vwénlns¹nzx¹ k€ivévu zén} |
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zv·nr qo Q Ëv nxsq kxn ëzq zv·rq qo tjivéj P ojljtt€} zv wvsy·jnzx¹ |
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vltj q zj |
n wsvxrvxz• |
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& − & + |
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ÌÒÆcÒÔ |
èjovi•nu kxn zénymvs•tqrq tj lkj rsjxxj ² zn y rvzvé€} kxn kné¡qt€ sn jz tj éjosq·t€} wé¹u€} q zn y rvzvé€} lkn kné ¡qt€ sn jz tj vltvp wé¹uvp ïqxsv zénymvs•tqrvk wnékvmv rsjxxj
éjktv S &Q u€ k€iqéjnu &Q xwvxvijuq zéq wé¹u€n tj rvzvé€} sn jz kné¡qt€ wvxsn ·nmv tj rj lvp qo ëzq} wé¹u€} k€iqéjnu wv vltvp
zv·rn qo S ïqxsv zénymvs•tqrvk kzvévmv rsjxxj éjktv Q& (Q − )S
S
u€ k€iqéjnu wé¹uyí tj rvzvévp sn jz lkn kné¡qt€ ojznu lkn zv·rq tj ëzvp wé¹uvp wvxsn ·nmv inénu léymyí wé¹uyí mln sn qz vltj kné¡qtj q zv·ry tj ëzvp wé¹uvp Ìi•nn ·qxsv zénymvs•tqrvk
éjktv |
Q(Q − )S (SQ + S − ) |
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P wjéjssns•t€} wé¹u€} lns¹z wsvxrvxz• tj P + wvsvxy
Çj lj¹ tvkj¹ wé¹uj¹ lvijks¹nz xzvs•rv ryxrvk tj xrvs•rv ·jxznp vtj lnsqzx¹ y n wévknlntt€uq wé¹u€uq Òjr rjr u€ wévkvlqu n•n Q wé¹u€} zv wvsy·jnu
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P + + (P + ) + (P + ) + + (P + Q) = |
(Q + )( P + Q) |
+ |
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·jxznp |
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èjovi•nu vréy |
tvxzq tj rsjxx€ wv |
·qxsy ojljtt€} zv·nr |
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sn |
j•q} tj ljttvp vréy |
tvxzq Ìltj vréy |
tvxz• j qunttv ljttj¹ |
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xvlné qz kxn ëzq zv·rq & & xvlné jz lkn zv·rq & & |
vlty zv·ry q |
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& |
² tq vltvp zv·rq Æxnmv wvsy·jnu + |
& & + & & + & = vr |
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éy |
tvxznp |
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Çj l€p wjéjssnsvméjuu vwénlns¹nzx¹ lkyu¹ wjéjuq wjéjs
sns•t€} wé¹u€} Ívëzvuy qunnu (& ) wjéjssnsvméjuuvk
Q+
j Éíi€n zéq |ns€} ·qxsj [ \ ] ylvksnzkvé¹í•qn tnéj kntxzkju Q + ≤ [ \ ] ≤ Q uvmyz i€z• xzvévtjuq zénymvs•tqrj Ívë
zvuy ·qxsv zénymvs•tqrvk x zjrquq xzvévtjuq éjktv &Q = &Q+
ïzvi€ tjpzq ·qxsv éjktvinléntt€} zénymvs•tqrvk ojunzqu ·zv wéq rj lvu ojljttvu vxtvkjtqq u€ qunnu Q éjktvinléntt€} zén
ymvs•tqrvk rj l€p éjo tvk€} dtj·qz q} vi•nn ·qxsv éjktv Q ïqxsv éjktvxzvévttq} zénymvs•tqrvk éjktv Q
i Ëju tjlv tjpzq rvsq·nxzkv zévnr tjzyéjs•t€} ·qxns [ \ ] zjrq} ·zv [ ≤ \ ≤ ] ≤ Q q [ + \ > ] Íyxz• ojljtv tnrvzvévn [ Òvmlj
\ wéqtqujnz otj·ntq¹ vz [ lv Q Çvmlj \ wévinmjnz otj·ntq¹ vz [ lv Q − [ + rj lvuy otj·ntqí \ xvvzknzxzkynz [ otj·ntqp ] ylvk
snzkvé¹í•q} tnéjkntxzkju \ ≤ ] < \ + [ ] ≤ Q cxsq n \ wéqtqujnz otj·ntq¹ vz Q − [ + lv Q zv ·qxsv xvvzknzxzkyí•q} otj·ntqp ] éjktv Q − \ + Æxnmv wéq ojljttvu [ wvsy·jnu
Q |
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[(Q − [ + ) + ∑ |
( Q − \ + ) = [Q − [ + [ |
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\= Q−[+ |
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-sjkj ,,
wjé (\ ]) zjrq} ·zv [ \ ] ylvksnzkvé¹íz wvxzjksntt€u yxsvkq¹u
Ìzxílj k€znrjnz ·zv vi•nn ·qxsv |
zénymvs•tqrvk ls¹ rvzvé€} |
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Q |
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[ |
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≤ [ ≤ Q q ≤ \ ] ≤ Q éjktv ∑ |
− [ + |
= Q (Q + ) Æ xqsy |
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[Q |
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[= |
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≥ Q + Ívëzvuy |
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w j xy•nxzkynz &Q+ zénymvs•tqrvk ls¹ rvzvé€} [ |
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kxnmv qunnu Q (Q + ) + Q(Q + )(Q + ) |
= Q(Q + )( Q + ) |
zénymvs•tq |
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rvk
Çvsq·nxzkv éjktvinléntt€} zénymvs•tqrvk x vxtvkjtqnu [ = N éjktv Q − N j x vxtvkjtqnu N + zv n Q − N Ívëzvuy vi•nn ·qxsv
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Q |
Q− |
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éjktvinléntt€} |
zénymvs•tqrvk éjktv ∑ ( Q − N) + ∑ ( Q − N) = Q |
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N= |
N= |
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Êxrsí·j¹ q} |
wvsy·jnu |
Q(Q + )( Q + ) − Q = |
Q(Q − )( Q − ) |
zén |
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ymvs•tqrvk
k êzj ojlj·j én¡jnzx¹ wvlvitv wénl€ly•np ïqxsv zénymvs•tqrvk
x ojljtt€u otj·ntqnu [ ≤ Q − éjktv Q[ − [ + [ j kxn} zénymvs•
Q− |
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− [ + [ = |
Q(Q + )(Q − ) |
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∑ |
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Q[ |
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[= |
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ïqxsv n zénymvs•tqrvk ls¹ rvzvé€} [ ≥ Q éjktv &Q+ j wvzvuy |
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kxnmv qunnu |
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Q(Q + )(Q − ) |
+ Q(Q + )(Q + ) = Q(Q + )( Q − ) |
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zénymvs•tqrvk ïqxsv éjktvinléntt€} zénymvs•tqrvk éjktv |
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Q− |
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Q− |
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∑ ( Q − N − ) + ∑ ( Q − N − ) = Q − Q + |
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N= |
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N= |
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j éjotvxzvévttq} ² |
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Q(Q + )( Q − ) − Q + Q − = |
(Q − )(Q − )( Q − ) |
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Ìivotj·qu xzvévt€ ·nz€én}ymvs•tqrj ·néno D E F G +H zné¹¹ vi•tvxzq uv tv x·qzjz• ·zv D ² tjqunt•¡j¹ qo xzvévt F ² wévzqkvwvsv tj¹ np xzvévtj q ·zv E < G Òvmlj D < E < G q D < F
Çévun zvmv zjr rjr ·nz€én}ymvs•tqr vwqxjt kvréym vréy tvxzq zv D + F = E + G Ìzxílj xsnlynz ·zv D + F > E Ívëzvuy wéq ojljtt€}
otj·ntq¹} D q E lsqtj F uv nz wéqtqujz• otj·ntq¹ vz E − D + lv
ÌÒÆcÒÔ
Q q lvs tv k€wvst¹z•x¹ tnéjkntxzkv E − D ≤ Q − Êzjr u€ lvrj
ojsq ·zv E ≤ D |
+ Q − |
q ·zv E − D + ≤ F ≤ Q Ìivotj·qu |
D + Q − |
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·néno V Òvmlj wéq ojljttvu otj·ntqq D qunnu |
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V |
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∑ (Q + D − E) = (V − D)(Q − V − ) |
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E=D+ |
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·nz€én}ymvs•tqrvk |
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Q = P Òvmlj |
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Íyxz• |
Q |
² ·nztvn |
·qxsv |
wéq D |
tn·nzt€} |
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D + Q − |
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D = N − qunnu V = |
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= P + N − q xsnlvkjzns•tv (P − N) |
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·nz€én}ymvs•tqrvk |
j |
wéq |
D |
·nzt€} |
D = N |
qunnu |
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D + Q − |
= P + N − q xsnlvkjzns•tv (P − N − )(P − |
N) ·nz€én} |
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V = |
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ymvs•tqrvk fyuuqéy¹ wv D wvsy·jnu ·zv vi•nn ·qxsv ·nz€én}ymvs• tqrvk éjktv
P |
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P |
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P(P − )( P − ) |
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Q(Q − )( Q − ) |
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∑ (P − N) + |
∑ (P − N)(P − N − ) = |
= |
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N= |
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N= |
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fsy·jp rvmlj Q tn·nztv éjxxujzéqkjnzx¹ jtjsvmq·tv |
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cxsq lvwyxrjz• ·zvi€ y ·nz€én}ymvs•tqrj i€sq vlqtjrvk€n xzv |
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évt€ |
zv |
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D |
E |
F |
G |
lvs |
t€ |
ylvksnzkvé¹z• |
xvvztv¡ntq¹u |
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D ≤ E ≤ G ≤ Q |
D ≤ F |
q D + F = E + G vzrylj |
xsnlynz ·zv |
E ≤ |
D + Q |
q |
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D + |
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E − D ≤ F ≤ Q cxsq wvsv |
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Q |
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qz• V = |
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zv ·qxsv ·nz€én}ymvs•tqrvk |
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wvstvp |
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rjéz€ uv tv |
k€tyz• |
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èvktv ujxzq iylyz k $ (& ) & |
= xsy·j¹} ² u€ k€iqéjnu |
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xwvxvijuq wvxsn ·nmv k€ |
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rjéz€ wvkzvé¹í•npx¹ ujxzq & |
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mvsvxvk un ly Q rjtlqljzjuq êzv uv tv xlnsjz• & Q−
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