Веленкин;Комбинаторика
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ÌÒÆcÒÔ
i ¬tjsvmq·tv wvsy·jnu & (& ) xwvxvivk |
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j ïqxsv zjrq} wénlxzjksntqp éjktv ·qxsy éjoiqntqp Q vlqtj |
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rvk€} ¡jévk tj tnwyxz€n méyww€ z n &Q− |
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= Q − Q + xwvxvijuq wénlxzjkqz• k kqln |
i ïqxsv Q uv tv &Q− |
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xyuu€ zén} |ns€} wvsv |
qzns•t€} xsjmjnu€} x·qzj¹ éjosq·t€uq |
wénlxzjksntq¹ vzsq·jí•qnx¹ wvé¹lrvu xsjmjnu€} cxsq lkj xsjmj
Q − nu€} éjkt€ zv vtq vwénlns¹íz q zénz•n zjrq} xsy·jnk iylnz
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wéq ·nztvu Q q wéq tn·nztvu Q ino y·nzj wvé¹lrj xsnlvkjtq¹
xsjmjnu€} Çévun zvmv nxsq Q lnsqzx¹ tj zv qunnzx¹ wénlxzjksn tqn k rvzvévu éjkt€ kxn zéq xsjmjnu€} wvëzvuy wéqlnzx¹ éjoiqéjz•xsy·j¹ Íéqunt¹¹ {véuysy krsí·ntqp q qxrsí·ntqp wvsy·jnu ·zv ·qxsv wénlxzjksntqp x wvwjétv éjosq·t€uq xsjmjnu€uq x y·nzvu wvé¹lrj q} xsnlvkjtq¹ k€éj jnzx¹ xsnlyí•quq {véuysjuq
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nxsq Q = N ± zv |
Q − Q + |
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nxsq Q = N ± zv |
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nxsq Q = N + zv |
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cxsq tn y·qz€kjz• wvé¹lvr xsjmjnu€} zv wvsy·qu k éjo unt•¡n wénlxzjksntqp Ënzéyltv wévknéqz• ·zv k€éj ntq¹ wvsy·jnu€n wéq
ëzvu ¹ks¹ízx¹ tn ·nu qt€u rjr |nsvp ·jxz•í Q − Q + wéq xv |
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vzknzxzkyí•q} otj·ntq¹} Q |
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j Òjr rjr = zv síivn éjosv |
ntqn uqssqvtj tj |
zéq utv qzns¹ qunnz kql ( D E )( D E )( D E ) |
mln kxn DL EL ² |
tnvzéq|jzns•t€n |ns€n ·qxsj q D + D + D = E + E + E = Òjr
rjr éjoiqkjnzx¹ tj tnvzéq|jzns•t€} |ns€} xsjmjnu€} & =
xwvxvijuq zv nxsq y·qz€kjz• wvé¹lvr utv qznsnp ·qxsv éjosv n tqp éjktv =
i Ívsy·ntt€n k wénl€ly•nu wytrzn éjosv ntq¹ éjxwjljízx¹ tj zéq rsjxxj sqiv kxn zéq utv qzns¹ xvkwjljíz sqiv lkj xvkwjljíz j zénzqp vzsq·jnzx¹ vz tq} sqiv kxn zéq éjosq·t€ Ínék€p rsjxx xvxzvqz qo vltvmv éjosv ntq¹ = Ëjplnu ·qxsv
éjosv ntqp kzvévmv rsjxxj cxsq xvkwjljíz wnék€n lkj utv qzns¹ zv wvsy·jnu ·zv D + D = E + E = Ëv yéjktntqn [ + \ = qunnz
én¡ntq¹ k |ns€} tnvzéq|jzns•t€} ·qxsj}
Òjr rjr síivn D uv tv rvuiqtqévkjz• x síi€u E wvsy·jnukjéqjtzvk ls¹ D E Ìlqt qo tq} j qunttv tjlv vziévxqz•
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rjr wéqkvl¹•qp r éjosv |
ntqí wnékvmv rsjxxj Ìxzjnzx¹ kjéqjtzvk |
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Çj l€p qo tq} wéqkvlqz r zénu éjosv |
ntq¹u k ojkqxquvxzq vz unxzj |
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rvzvévn ojtqujnz zénzqp utv |
qzns• dtj·qz kzvévp rsjxx xvxzvqz qo |
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ntqp cxsq tn y·qz€kjz• wvé¹lrj xvutv qznsnp wvsy·jnu |
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éjosv |
ntqp |
Ëjrvtn| ·qxsv éjosv |
ntqp zénz•nmv rsjxxj éjktv |
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= Ìtq éjxwjljízx¹ tj méyww€ vzsq·jí•qnx¹ sq¡• |
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wvé¹lrvu xvutv |
qznsnp q xvxzv¹•qn qo éjosv ntqp rj |
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méyww iylnz Ívëzvuy nxsq tn y·qz€kjz• wvé¹lrj xvutv |
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qznsnp u€ qunnu éjosv ntqp |
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lnsqznsnp éjxxuvzéqu |
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+ S + + SQN) cxsq éjxré€z• k |
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xrvirq zv wvsy·qu xyuuy k rvzvéyí rj l€p lnsqzns• k}vlqz évktv |
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vlqt éjo Ív {véuysn ls¹ xyuu€ mnvunzéq·nxrvp wévménxxqq wvsy·j |
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nu ·zv ëzj xyuuj éjktj |
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j ïqxsj ; q < vzsq·jízx¹ |
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vz léymj utv qzns¹uq $D |
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%E &F 'G ² rj l€p qo ëzq} ·nz€én} utv qznsnp k}vlqz k vltv qo ·qxns q tn k}vlqz k léymvn Òjr rjr utv qzns¹ uv tv éjxwénlnsqz•
un ly lkyu¹ ·qxsjuq = xwvxvijuq zv ojlj·j qunnz én¡ntqp cxsq tn y·qz€kjz• wvé¹lvr ·qxns zv qunnu én¡ntqp
i Êxrvu€n ·qxsj quníz kql *. q */ mln . q / ² lnsqznsq ·qxsj 1 = $D%E&F'G êzv ·qxsv qunnz τ(1) = (D + )(E + )(F + )(G + )
lnsqznsnp Ívëzvuy nxsq tn éjosq·jz• wjé€ (*. */) q (*/ *.) zv
. q / uv __tv k€iéjz• &τ(1) = &τ(1)+ xwvxvijuq j nxsq éjosq·jz• zjrqn wjé€ zv $τ(1) = (τ(1)) xwvxvijuq
èjxxzjkqu wénlunz€ k tnrvzvévu wvé¹lrn q vzljlqu wnékvuy ·nsvknry wnék€n Q wénlunzvk kzvévuy ² kzvé€n Q wénlunzvk q wvx snltnuy ² vxzjk¡qnx¹ Q wénlunzvk Ívxrvs•ry wvé¹lvr ësnuntzvk k
méywwj} tn qméjnz évsq wvsy·jnu |
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xwvxvivk éjolnsj Ív |
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xsnlvkjzns•t€p k€ivé wéqkvlqz r zvuy |
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n vzknzy tv k kqln & Q&Q |
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3(P Q S) = (P + Q + S)
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ÌÒÆcÒÔ |
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Çj lvp wnénxzjtvkrn xvvzknzxzkynz xkvp xwvxvi éjxxzjtvkrq rtqm
Òv·tv zjr |
n rjr k wénl€ly•np ojlj·n wvsy·jnu ·zv ·qxsv |
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xwvxvivk éjktv |
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Æ Q utv nxzkn Q wjé (D E) Ìztv¡ntqn ² wvlutv nxzkv ëzvmv |
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utv nxzkj Íéq |
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wjé kqlj (D D) iylnz Q vi~nlqtntt€} wjé |
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méyww qo = kvouv t€} Ívëzvuy qunnu & = xwvxvivk
éjxxzjtvkrq
i -éyww xzjsv xnu• wvëzvuy qunnu & = xwvxvi
j ¬tjsvmq·tv wénl€ly•np ojlj·n qunnu = méyww€ q
& xwvxvivk
i Ënz xwvxvivk xquunzéq·tvn ·nétvuy wvsí ² insvn
k & q & wvsnp xzjsv klkvn unt•¡n
j 3( ) xwvxvivk Ëjlv tj vltvp wvsvkqtn lvxrq éjxxzj
kqz• ins€} q ·nét€} ¡j¡nr tj ·nét€} wvs¹} wvs¹ iylyz wyxz€uq q vzéjoqz• éjxxzjtvkry tj léymyí wvsvkqty lvxrq
i & xwvxvivk Ëjlv tj vltvp wvsvkqtn lvxrq k€iéjz•
wvsnp qo q wvxzjkqz• tj rj lvn ¡j¡ry wévqokvs•tvmv |knzj j tj kzvévp wvsvkqtn ojt¹z• xquunzéq·t€n wvs¹ ¡j¡rjuq wévzqkvwvsv tvmv |knzj
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èjovi•nu xtj·jsj qmévrvk rj lvp xzéjt€ tj ywvé¹lv·ntt€n wjé€ bs¹ rj lvp xzéjt€ ëzv uv tv xlnsjz• = xwvxvijuq wv
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wvsy·jnu Q ( Q) lvwyxzqu€} wnénxzjtvkvr
Ìivotj·qu tnwyxzyí xvkvrywtvxz• ins€} ¡jévk iyrkvp È j ·nét€} ¡jévk iyrkvp ï Êo yxsvkq¹ ojlj·q xsnlynz ·zv ¡jé€ éjx wvsjmjízx¹ wv vltvp qo x}nu ïÈïÈ ïÈ qsq ÈïÈï Èï wéq·nu k
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U− xwvxvivk Òv·tv zjr |
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qunnu &Q− xwvxvivk j kxnmv &P− &Q− |
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k€kvlqu ·zv U rvtzjrzvk iylyz k &P− &Q− |
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Æ rj lvp lvwyxzquvp wnénxzjtvkrn rjr jtmsq·jtn zjr q {éjt |yo€ xzv¹z méywwjuq xvxzv¹•quq tn untnn ·nu qo lky} ·nsvknr Íéq
ëzvu ·qxsv méyww {éjt|yovk vzsq·jnzx¹ vz ·qxsj méyww jtmsq·jt tn |
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ivsnn ·nu tj Ívlx·qzjnu xrvs•rquq xwvxvijuq uv tv éjoiqz• Q |
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jtmsq·jt tj S ywvé¹lv·ntt€} méyww zjr ·zvi€ rj |
lj¹ méywwj xvlné |
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jsj tn untnn lky} ·nsvknr bs¹ ëzvmv tjlv éjxwvsv qz• q} k tnrv |
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S− xwvxvijuq éjoiqz• tj S méyww tn |
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zvévu wvé¹lrn Q xwvxvivk q &Q−S− |
S− xwvxvivk Òv·tv zjr |
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untnn ·nsvknr k méywwn Æxnmv qunnu Q &Q−S− |
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V méyww |
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êzq xwvxvi€ uv tv rvuiqtqévkjz• léym x léymvu xsnlyí•quq unzvljuq ¬ ² méywwj jtmsq·jt Ð ² {éjt|yovk ¬Ð¬Ð Ь ¬Ð¬Ð ¬Ð ЬЬ Ь ЬЬ Ð dtj·qz vi•nn ·qxsv xwvxvivk k€éj jnzx¹ {véuysvp
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èjxré€kj¹ xrvirq k {véuysn k ojlj·n wvsy·jnu zvz |
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cxsq k xsvkv k}vlqz N tysnp zv k tnu Q − N tn tysnp rvzvé€n |
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kxzjkqz• vlqtv·t€n tysq k€iéjk unxzj ls¹ tq} &Q−N+ xwvxvijuq |
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Ìzknz wvsy·jnzx¹ k kqln Q + Q− & + Q− |
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tv éjolnsqz• wvwvsju |
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xwvxvijuq Òjr rjr ëzq éjoiqntq¹ uv tv |
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lkyu¹ xwvxvijuq rvuiqtqévkjz• léym x léymvu wvsy·jnu ( ) xwv xvivk éjoiqntq¹ béymj¹ svmqrj wéqkvlqz r vzknzy & &
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tv éjolnsqz• xwvxvijuq Æxnmv wvsy·jnu & = xwvxvivk |
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éjxwénlnsqu ¹isvr Òjr rjr rj l€p wvsy·jnz tn |
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wv x}nun |
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+ + tjlv k€iéjz• ls¹ kzvévmv {éyrzj qo & |
xwvxvivk Òjr |
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rjr lkj ·nsvknrj quníz wvévkty ¹isvr zv ·qxsv wnénxzjtvkvr sílnp éjktv 3( ) = Íéq x}nun + + tjlv k€iéjz• vlqt {éyrz qo
ls¹ wnékvmv q vlqt {éyrz qo vxzjk¡q}x¹ ls¹ kzvévmv dlnx• zjr n 3( ) wnénxzjtvkrq sílnp
Òv·tv zjr |
n éjxxujzéqkjízx¹ vxzjs•t€n x}nu€ Æxnmv wvsy·jnu |
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3 & + 3( )& + 3( )& & + 3 & & + & & = |
xwvxvivk |
éjxwénln |
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tv wénl€ly•np ojlj·n wvsy·jnu |
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xwvxvivk éjxwénlnsntq¹ |
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j 3( ) i 3( ) k & & 3( ) |
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nt qunz• wv vltvp rjézn rj |
lvp ujxzq zv ls¹ rj |
lvp ujxzq |
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wvsy·jnu wnénxzjtvkry qo rjéz wvxrvs•ry wnénxzjtvkrq ujxznp tn ojkqx¹z léym vz léymj zv wv wéjkqsy wévqoknlntq¹ wvsy·jnu ( ) xwvxvivk Æ zénz•nu xsy·jn vlqt qmévr k€iqéjnu€p xwvxv
ijuq uv nz k€iéjz• wv rjézn rj lvp ujxzq xwvxvijuq Ívxsn ëzvmv vxzjk¡qnx¹ rjéz rj lvp ujxzq uv tv éjolnsqz• tj méyww€
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sv nt zv ls¹ rj lvp qo Q + syo tjlv xlnsjz• k€ivé ² rsjxz• k
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