Веленкин;Комбинаторика
.pdf-sjkj ,9 è¬dÈÊcËʉ
Æ ëzvp msjkn u€ éjxxuvzéqu éjot€n kjéqjtz€ y n kxzén ·jk¡npx¹ ojlj·q v éjoiqntqq ·qxsj tj xsjmjnu€n Ç ëzvp ojlj·n xkvl¹zx¹ utvmqn léymqn wéq·nu olnx• kvotqrjíz éjot€n vxv inttvxzq k ojkqxquvxzq vz vméjtq·ntqp tjrsjl€kjnu€} tj knsq·qty qsq ·qxsv xsjmjnu€}
djlj·j v tjrsnprn ujévr
dj wnénx€sry ijtlnévsq tjlv ywsjzqz• éyisnp tjrsn qkj¹ tj tnn ujérq Ëj wv·zn nxz• wv vltvuy kqly ujévr lvxzvqtxzkvu k q éyisnp ojzv k tnvméjtq·nttvu rvsq·nxzkn frvs•rquq xwvxvijuq uv tv vwsjzqz• wnénx€s ry ijtlnévsq nxsq lkj xwvxvij vzsq·jí•qnx¹ tvuqtjsvu qsq wvé¹lrvu tjrsnqkjtq¹ ujévr x·qzjízx¹ éjosq·t€uq ujérq
tjrsnqkjízx¹ k vlqt é¹l "
Ìivotj·qu ·néno )(1) ·qxsv xwvxvivk rvzvé€uq uv tv
tjrsnqz• ujérq lvxzvqtxzkvu k q olnx• q ljsnn éyisnp
zjr ·zvi€ vi•j¹ xzvquvxz• ëzq} ujévr éjkt¹sjx• 1 bs¹ ujs€} otj·ntqp 1 snmrv tjpzq )(1) tnwvxénlxzknttv Íévxzvp
wvlx·nz wvrjo€kjnz ·zv )( ) = tn tjrsnqkjz• tq vltvp ujérq )( ) = )( ) = )( ) = rjr tq tjrsnqkjz• ujérq yrj
ojttvmv k€¡n lvxzvqtxzkj xyuu€ k qsq tn wvsy·qzx¹ bjsnn )( ) = tjrsnqkjnzx¹ vltj ujérj lvxzvqtxzkvu k
)( ) = )( ) = )( ) = )( ) = )( ) = Ënwvxénlxzkntt€p wvlx·nz wvrjo€kjnz zjr n ·zv )( ) = ujérq
bjs•¡n tnwvxénlxzkntt€p wvlx·nz xzjtvkqzx¹ ojzéyltqzns• t€u tv vt y n q tn ty nt bvxzjzv·tv ojunzqz• ·zv ls¹ síivmv 1 knétv énryééntztvn xvvztv¡ntqn
)(1) = )(1 − ) + )(1 − ) + )(1 − ) |
|
Æ xjuvu lnsn wyxz• qunnzx¹ tnrvzvé€p xwvxvi tjrsnprq ujévr x vi•np xzvquvxz•í 1 q wyxz• wvxsnltnp tjrsnntj ujérj xzv
quvxz•í k éyis¹ Òvmlj kxn vxzjs•t€n ujérq kunxzn xzv¹z 1 − Ëjvivévz wéqxvnlqt¹¹ k rvtn| síivp rvuiqtj|qq uj
évr vi•np xzvquvxz•í 1 − vlty éyisnkyí ujéry wvsy·jnu
rvuiqtj|qí ujévr xzvquvxz•í 1 Íéq ëzvu qo éjot€} rvuiq tj|qp xzvquvxz•í 1 − wvsy·jzx¹ éjot€n rvuiqtj|qq xzvq
uvxz•í 1 Êzjr ·qxsv qxrvu€} rvuiqtj|qp mln wvxsnltnp tjrsnntj éyisnkj¹ ujérj éjktv )(1 − )
Òv·tv zjr n lvrjo€kjnzx¹ ·zv ·qxsv rvuiqtj|qp vrjt ·qkjí•q}x¹ tj éyisnkyí ujéry éjktv )(1 − ) j tj éyi
snkyí ujéry vrjt·qkjízx¹ )(1 − ) rvuiqtj|qp Ívxrvs•ry
djlj·j v tjrsnprn ujévr |
|
|
|
síij¹ rvuiqtj|q¹ vrjt·qkjnzx¹ tj ujéry vltvmv qo yrjojtt€} zqwvk zv wv wéjkqsy xyuu€ wvsy·jnu xvvztv¡ntqn
¬ znwné• y n snmrv tjpzq xrvs•rquq xwvxvijuq uv tv wvsy·qz• xyuuy z n k€·qxsqz• )( ) Ív {véuysn
)( ) = )( ) + )( ) + )( )
dtj·ntqn )( ) u€ y n otjnu ² vtv éjktv ¬ r )( ) q )( ) u€ n•n éjo wéquntqu {véuysy
)( ) = )( ) + )( ) + )( ) = + + = )( ) = )( ) + )( ) + )( ) = + + =
dtj·qz )( ) = + + =
Êzjr xyuuy uv tv wvsy·qz• xwvxvijuq Ævz vtq
+ + + + + + + + + + ++ + + + + + + + +
Ìzunzqu ·zv otj·ntq¹ )(1) ls¹ 1 =
uv tv i€sv wvsy·qz• qtj·n tn wévkvl¹ tnwvxénlxzknttvp wévknérq bnsv k zvu ·zv wéq 1 < qunnu )(1) = wvxrvs•ry
vzéq|jzns•tyí xyuuy tns•o¹ ywsjzqz• tjrsnqkj¹ tnvzéq|j
zns•tvn rvsq·nxzkv ujévr Æ zv n kénu¹ rjr u€ kqlnsq
)( ) = Ívëzvuy )( ) = )(− ) + )(− ) + )(− ) =
Òv·tv zjr n wvsy·jnu otj·ntqn )( ) = )( ) = ¬ ls¹
1 = qunnu )( ) = )( ) + )(− ) + )(− ) =
èjoviéjttj¹ ojlj·j ¹ks¹nzx¹ ·jxzt€u xsy·jnu xsnlyí•np vi•np ojlj·q
Êunízx¹ ujérq lvxzvqtxzkvu k Q Q QN éyisnp kxn ·qxsj QL éjosq·t€ j ojwjx ujévr tnvméjtq·nt frvs•rquq
xwvxvijuq tjrsnqkj¹ ujérq k é¹l uv tv vwsjzqz• x q} wvuv••í xyuuy k 1 éyisnp nxsq lkj xwvxvij vzsq·jí•qnx¹
wvé¹lrvu xsnlvkjtq¹ éjosq·t€} ujévr x·qzjízx¹ éjot€uq" Æ ëzvu xsy·jn ·qxsv )(1) xwvxvivk ylvksnzkvé¹nz xvvztv¡n
tqí
)(1) = )(1 − Q ) + )(1 − Q ) + + )(1 − QN) |
|
Íéq ëzvu )(1) = nxsq 1 < q )( ) = f wvuv••í xvvztv ¡ntq¹ uv tv tjpzq )(1) ls¹ síivmv 1 wvxsnlvkjzns•tv k€·qxs¹¹ )( ) )( ) )(1 − )
Òjrvp xwvxvi én¡ntq¹ ojlj· wéq rvzvévu xéjoy tn ljnzx¹ vrvt·jzns•tvp {véuys€ ls¹ vzknzj j yrjo€kjnzx¹ sq¡• wév |nxx wvokvs¹í•qp xkvlqz• ojlj·y rv kxn unt•¡qu q unt•¡qu ·qxsvk€u ljtt€u kxzén·jnzx¹ k rvuiqtjzvéqrn v·nt• ·jxzv
|
-sjkj ,9 è¬dÈÊcËʉ |
|
|
èjkntxzkj kqlj tjo€kjíz énryééntzt€uq {véuysjuq Ív lévitnn u€ tj ëzvu vxzjtvkqux¹ k msjkn 9,,
èjoiqntqn ·qxns tj xsjmjnu€n
Æojlj·n v tjrsnprn ujévr u€ lvs t€ i€sq k€iqéjz• ujérq sq¡• vméjtq·nttvmv ·qxsj kqlvk èjxxuvzéqu vlqt ·jxzt€p xsy·jp ëzvp ojlj·q
Ëj wv·zn qunnzx¹ wv vltvuy kqly ujévr rj lvmv tvuqtjsjk éyis¹ q z l nxznxzknttv k lvxzjzv·tvu rvsq·nxzkn frvs•rquq éjot€uq xwvxvijuq uv tv k ëzvu xsy·jn tjrsnqz• k é¹l ujérq tj xyuuy k 1 éyisnp"
èjot€uq vw¹z• x·qzjízx¹ xwvxvi€ vzsq·jí•qnx¹ rvsq·n xzkvu ujévr qsq wvé¹lrvu tjrsnqkjtq¹ éjosq·t€} ujévr êzy ojlj·y uv tv x{véuysqévkjz• qtj·n
frvs•rquq xwvxvijuq uv tv wénlxzjkqz• ·qxsv 1 k kqln xyuu€ tjzyéjs•t€} xsjmjnu€} nxsq tnz tqrjrq} vméjtq·n tqp tq tj xjuq xsjmjnu€n tq tj q} ·qxsv j lkj éjoiqntq¹ vzsq·jí•q}x¹ wvé¹lrvu xsnlvkjtq¹ éjosq·t€} xsjmjnu€} x·qzjízx¹ éjosq·t€uq"
Îtv nxzkv kxn} zjrq} éjoiqntqp ·qxsj 1 uv tv éjoiqz• tj rsjxx€ vztnx¹ k N p rsjxx éjoiqntq¹ x N xsjmjnu€uq Çj lvuy zjrvuy éjoiqntqí vzkn·jnz éjxrsjlrj 1 vlqtjrvk€} ¡jévk k N ¹•qrvk wéq rvzvévp tq vlqt qo ¹•qrvk tn wyxz
¬ ·qxsv zjrq} éjxrsjlvr rjr u€ otjnu xu w éjktv
&N1−−
ïzvi€ tjpzq vi•nn ·qxsv éjoiqntqp tjlv wévxyuuqévkjz• |
||
|
+ & |
+ + &1− |
wvsy·ntt€n vzknz€ wv N vz lv 1 &1− |
− |
1− Ëv |
ëzj xyuuj éjktj 1− dtj·qz xy•nxzkynz 1− xwvxvivk éjosv
qz• ·qxsv 1 tj tjzyéjs•t€n xsjmjnu€n Ëjwéquné ·qxsv uv tv éjoiqz• tj xsjmjnu€n − = éjosq·t€uq xwvxvijuq
= |
= + + |
= + + + |
= + |
= + + |
= + + + |
= + |
= + + |
= + + + |
= + |
= + + |
= + + + |
= + |
= + + |
= + + + + |
|
= + + |
|
Énmrv wvt¹z• vzrylj ko¹svx• ·qxsv 1− Ìtv xvvzknzxzkynz xsn lyí•np x}nun djwq¡nu k é¹l 1 nlqtq| În ly tquq iylnz 1 −
wévun yzvr Æ rj l€p qo tq} uv tv wvxzjkqz• otjr © ª j uv tv
tn xzjkqz• În ly otjrjuq © ª wvsy·jzx¹ rjr éjo xsjmjnu€n Ëjwéq uné = + + wvsy·qzx¹ rjr + +
•nzvt€ k un¡rn |
|
|
|
cxsq lvwyxzqz• tysnk€n xsjmjnu€n tv ojljz• ·qxsv xsjmj
nu€} éjkt€u N zv wvsy·qu ojlj·y v éjxrsjlrn 1 ¡jévk wv
N−
N ¹•qrju rvzvéj¹ én¡jnzx¹ & +N− xwvxvijuq
djlj·q
frvs•rquq xwvxvijuq uv tv tjrsnqz• tj rvtknéz k vlty sqtqí ujérq tj éyisnp qxwvs•oy¹ ujérq lvxzvqtxzkvu k q éyisnp éjxwvsv ntq¹ vzsq·jí•qnx¹ wvé¹lrvu ujévr éjxxujzéqkj ízx¹ rjr éjosq·t€n ·qxsv ujévr tn vméjtq·ntv "
frvs•rquq xwvxvijuq uv tv wvsy·qz• wv kvx•uq éjot€u wénlunzju v|ntrq qsq zjr ·zvi€ q} xyuuj éjkt¹sjx• "
ïqxsj vz lv Q k€wqxjt€ wvlé¹l zjr ·zv wvsy·qsjx• wvxsnlvkjzns•tvxz• |q{é bvrj qzn ·zv
·qxsv kxn} |q{é ëzvp wvxsnlvkjzns•tvxzq éjktv ·qxsy tysnp k wvxsnlvkjzns•tvxzq Q+
•nzvt€ k un¡rn
fjuvxzv¹zns•tvn éjokqzqn rvuiqtjzvéqrq tj·jsvx• x wnén wqxrq un ly Íjxrjsnu q Ðnéuj v tnrvzvé€} ojlj·j} xk¹ojt t€} x qmévp k rvxzq Ìltvp qo ëzq} ojlj· i€s wvlx·nz ·qxsj xwvxvivk wvsy·qz• xyuuy 1 v·rvk wéq iévxjtqq Q rvxznp qsq wéq Q réjztvu iévxjtqq vltvp rvxzq ² ëzv vltv q zv n wvzvuy ·zv rvxzq uv tv x·qzjz• wnéntyunévkjtt€uq êzj ojlj·j isqorj r éjxxuvzénttvp tjuq ojlj·n v tjrsnqkjtqq ujévr qsq éjoiqntqq ·qxsj tj xsjmjnu€n ² k€wjlntqn vwén lnsnttvmv ·qxsj v·rvk tj rvxzq uv tv éjxxujzéqkjz• rjr tjrsnqkjtqn v·nénltvp ujérq tvuqtjsvu vz lv éyisnp qsq lvijksntqn k xyuuy v·nénltvmv xsjmjnuvmv Ëv nxz• xy•nxzknt tvn vzsq·qn ² ·qxsv ©ujévrª znwné• {qrxqévkjtv ² iévxjnzx¹ Q rvxznp
èjxxuvzéqu znwné• q u€ lj n tn ëzy j ivsnn vi•yí ojlj·y
Êunnzx¹ un¡vr k rvzvévu sn qz P ojtyunévkjtt€} nzvtvk ² tj tq} tjwqxjt€ ·qxsj P − Êo un¡rj
k€zjxrqkjíz v·nénltvp nzvt ojwqx€kjíz nmv tvuné ojznu nzvt rsjlyz viéjztv k un¡vr Òjr wvsy·jíz wvxsnlvkj
zns•tvxz• qo Q ·qxns rj lvn qo rvzvé€} ojrsí·ntv k wénln sj} vz lv P − Òéniynzx¹ yotjz• xrvs•rv qo wvsy·jí•q}
x¹ zjrqu viéjovu éjosq·t€} wvxsnlvkjzns•tvxznp quníz xyuuy 1
êzv rvsq·nxzkv vivotj·qu &P(Q 1) |
|
èjxxuvzéqu wvxsnltqp |
Q p qo k€zj•ntt€} nzvtvk Ëj |
tnu uv nz i€z• síivn qo |
P ·qxns P − cxsq tj |
tnu xzvqz zv wnék€n Q − |
nzvtvk lvs t€ ljkjz• xyuuy 1 |
¬ rvsq·nxzkv wvxsnlvkjzns•tvxznp lsqt€ Q − x xyuuvp 1 éjktv &P(Q − 1) ·qxsv P éjosq·t€} nzvtvk u€ x·qzjnu
-sjkj ,9 è¬dÈÊcËʉ
{qrxqévkjtt€u cxsq tj wvxsnltnu nzvtn xzvqz zv wnék€n |
|
Q − nzvtvk lvs |
t€ ljkjz• xyuuy 1 − j zjrq} wvxsnlvkj |
zns•tvxznp qunnzx¹ |
&P(Q − 1 − ) Ævvi•n ·qxsv wvxsnlvkj |
zns•tvxznp lsqt€ Q x xyuuvp 1 q wvxsnltqu ·sntvu N éjktv &P(Q − 1 − N) Ívëzvuy wv wéjkqsy xyuu€ vi•nn ·qxsv wv
xsnlvkjzns•tvxznp lsqt€ Q x xyuuvp 1 éjktv
&P(Q 1) = &P(Q − 1) + &P(Q − 1 − ) +
+ &P(Q − 1 − ) + + &P(Q − 1 − P + )
u€ wvsjmjnu &P(N 1) = nxsq N <
Ívs•oy¹x• ëzqu éjkntxzkvu uv tv xknxzq ojlj·y v unzjtqq Q rvxznp r ojlj·n v unzjtqq Q − rvxznp nn ² r ojlj·n v
unzjtqq Q − rvxznp q wévlvs jz• ëzvz wév|nxx lv zn} wvé wvrj tn vxzjtnzx¹ tq vltvp rvxzq ¬ ls¹ Q = kxn v·nkqltv
² k€iéjx€kj¹ rvxznp u€ uv nu nlqtxzkntt€u viéjovu
wvsy·qz• v·rvk q tqrjrvmv qtvmv rvsq·nxzkj v·rvk wvsy·qz• tn uv nu dtj·qz &P( ) = q &P( 1) = nxsq 1 <
2 jéq{unzq·nxrqp zénymvs•tqr
Íéq P = k un¡rn nxz• sq¡• lkj nzvtj tj tq} tjwqxjt€
·qxsj q Ívëzvuy wvsy·jízx¹ wvxsnlvkjzns•tvxzq lsqt€ Q qo tysnp q nlqtq| j xyuuj ·qxns wvxsnlvkjzns•tvxzq éjktj ·qxsy nlqtq| k tnp Ëv rvsq·nxzkv wvxsnlvkjzns•tvxznp qo
1 nlqtq| q Q − 1 tysnp éjktv 3(1 Q − 1) z n &1 Êt€uq |
|||
|
|
|
Q |
xsvkjuq & |
(Q 1) = &1 Ðvéuysj wéq P = wénkéj•jnzx¹ |
||
|
Q |
|
+ &1− |
k y n otjrvuvn tju xvvztv¡ntqn &1 = &1 |
|||
|
Q |
Q− |
Q− |
Æ w x wvuv••í ëzvmv xvvztv¡ntq¹ i€s wvxzévnt jéq{ unzq·nxrqp zénymvs•tqr Òv·tv zjr n xvvztv¡ntqn wvokv
s¹nz wvxzévqz• P jéq{unzq·nxrqp zénymvs•tqr cmv ësnuntz€ wéq ·nztvu P = N éjxwvsjmjízx¹ zjr ·zv wvl ·qxsjuq v·n
énltvp xzévrq tj}vl¹zx¹ wévun yzrq un ly ·qxsjuq xsnlyí •np xzévrq Íéq ëzvu rj lvn ·qxsv éjktv xyuun N ·qxns
wénl€ly•np xzévrq N ·qxns tj}vl¹•q}x¹ xsnkj vz tnmv q N ·qxns tj}vl¹•q}x¹ xwéjkj cxsq n P = N + ² tn·nztvn
·qxsv zv ·qxsj wq¡yzx¹ léym tjl léymvu q rj lvn ·qxsv éjktv xyuun tj}vl¹•nmvx¹ tjl tqu ·qxsj N ·qxns éjxwvsv ntt€} k wénl€ly•np xzévrn xsnkj vz tnmv q N ·qxns éjxwvsv ntt€} k wénl€ly•np xzévrn xwéjkj vz tnmv Æ vivq} xsy·j¹} xyuuj éjxwvsjmjnzx¹ xquunzéq·tv vztvxqzns•tv xsjmjnu€} cxsq xsnkj qsq xwéjkj vz qxrvuvmv ·qxsj k wénl€ly•np xzévrn ·qxns
2 jéq{unzq·nxrqp zénymvs•tqr |
|
tn }kjzjnz zv tnlvxzjí•qn xsjmjnu€n wvsjmjízx¹ éjkt€uq tysí
Æ€wq¡nu P jéq{unzq·nxrqn zénymvs•tqrq ls¹ P = q ls¹ P =
P = & (Q 1) = & (3 − ) + (Q − 1 − ) + (Q − 1 − )
& ( 1) |
|
|
|
|
|
|
|
|
|
|
|
|
& ( 1) |
|
|
|
|
|
|
|
|
|
|
|
|
& ( 1) |
|
|
|
|
|
|
|
|
|
|
|
|
& ( 1) |
|
|
|
|
|
|
|
|
|
|
|
|
& ( 1) |
|
|
|
|
|
|
|
|
|
|
|
|
& ( 1) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||
|
|
|||||||||||
P = & (Q 1) = & (3 − ) + (Q − 1 − ) + + (Q − − ) |
||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
& ( 1) |
|
|
|
|
|
|
|
|
|
|
|
|
& ( 1) |
|
|
|
|
|
|
|
|
||||
& ( 1) |
|
|
|
|
|
|||||||
& ( 1) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||
|
|
|||||||||||
Çj lj¹ xzévrj ëzq} zjisq| tj·qtjnzx¹ x êzv wvrjo€kjnz |
||||||||||||
·zv xyuuy |
ljnz sq¡• |
vltj |
wvxsnlvkjzns•tvxz• |
nzvtvk ² |
||||||||
xvxzv¹•j¹ qo Q tysnp fsnlyí•nn oj ·qxsv éjktv Q z n rvsq·nxzky k€zjxrqkjnu€} nzvtvk ² xyuuy uv tv wvsy ·qz• Q xwvxvijuq wvxzjkqk tj vltv qo Q unxz j vxzjs•t€n unxzj ojwvstqk tys¹uq Ævvi•n nxsq tyunéj|qí xzévr q
ësnuntzvk k xzévrn tj·jz• x tys¹ zv 1 n ·qxsv k Q p xzévrn éjktv &P(Q 1) ² ·qxsy wvxsnlvkjzns•tvxznp lsqt€ Q qo ·qxns
vz lv P − x xyuuvp ·qxns 1 Ëv ·qxsj vz lv P − ²
ëzv |q{é€ P q·tvp xqxznu€ x·qxsntq¹ Ívëzvuy zjryí wvxsn
lvkjzns•tvxz• uv tv éjxxujzéqkjz• rjr Q otj·tvn ·qxsv k P q·tvp xqxznun x·qxsntq¹ dtj·qz &P(Q 1) éjktv rvsq·nxzky
Q otj·t€} ·qxns k P q·tvp xqxznun x·qxsntq¹ y rvzvé€} xyuuj |q{é éjktj 1
Íéq ëzvu znéuqt ©Q otj·t€nª u€ wvtqujnu k ¡qévrvu xu€xsn lvwyxrj¹ q ·qxsj tj·qtjí•qnx¹ x vltvmv qsq tnxrvs• rq} tysnp Ëjwéquné éjxxujzéqkjnzx¹ rjr ¡nxzqotj· tvn ·qxsv xyuuj |q{é rvzvévmv éjktj
|
-sjkj ,9 è¬dÈÊcËʉ |
|
|
Ëj vi€·t€} qméjs•t€} rvxz¹} v·rq qlyz vz lv j tn vz lv Ívëzvuy ·zvi€ yotjz• xrvs•rquq xwvxvijuq uv tv wvsy·qz• 1 v·rvk k€iévxqk Q rvxznp tjlv xtj·jsj k€·nxz• wv vz ·qxns tj rj lvp méjtq Æ énoys•zjzn xyuuj v·rvk
yunt•¡qzx¹ tj Q Êt€uq xsvkjuq ·qxsv xwvxvivk wvsy·qz• 1 v·rvk wéq iévxjtqq Q rvxznp éjktv & (Q 1 − Q) Ëjwéquné
nxsq ·qxsv rvxznp éjktv j xyuuj v·rvk ² zv ·qxsv xwvxvivk éjktv & ( ) Ív zjisq|n ls¹ P = tj}vlqu ·zv vtv
éjktv
f·jxzsqk€n zévssnpiyxt€n iqsnz€
c•n tn zjr ljktv k jkzviyxj} q zévssnpiyxj} wjxxj qé€ wéq rj lvp wvnolrn wvsy·jsq y rvtlyrzvéj qsq k rjxxn iqsnz x ¡nxzqotj·t€u tvunévu tj t€tn¡tq} vltvéjovk€} zjsvtj} zv n nxz• tvuné tv vt lnx¹zqotj·t€p Ënrvzvé€n sílq x·qzjsq zjrvp iqsnz ©x·jxzsqk€uª nxsq k nmv tvunén xyuuj |q{é k wnékvp wvsvkqtn tvunéj éjkt¹sjx• xyuun |q{é kzvévp wvsvkqt€ tvunéj Ëjwéquné iqsnz x tvunévu qunnz éjkt€n xyuu€ wnék€} q wvxsnltq} zén} |q{é
|
wvëzvuy vt x·jxzsqk€p ¬ tn |
rvzvé€n x·qzjsq iqsnz x·jxzsqk€u nxsq xyuuj |q{é xzv¹•q} |
|
tj ·nzt€} unxzj} i€sj éjktj xyuun |q{é xzv¹•q} tj tn·nz |
|
t€} unxzj} Ëjwéquné |
iqsnz x·qzjnzx¹ x·jxzsqk€u |
² Ëv nxsq u€ wnék€n zéq |q{é€ tvunéj wnénxzjkqu tj n n q n unxzj vz tj·jsj j wvxsnltqn zéq |q{é€ ² tj n n q n unxzj zv x·jxzsqk€p iqsnz wnékvmv zqwj wnénplnz k x·jxzsqk€p iqsnz kzvévmv zqwjuv tv wévlnsjz• q viéjztyí vwnéj|qí Ívëzvuy q} rvsq·n xzkj éjkt€ frvs•rv n xy•nxzkynz ©x·jxzsqk€}ª tvunévk vz
lv "
ïzvi€ vzknzqz• tj ëzvz kvwévx xtj·jsj tjplnu xrvs•rv zén}otj·t€} ·qxns qunnz ljttyí xyuuy |q{é 1 wéq ëzvu u€ vztvxqu r zén}otj·t€u q ·qxsj kqlj q lj n dlnx•
·qxsv xsjmjnu€} xyuuj éjktj 1 j xsjmjnu€n ² vz lv wvëzvuy zjrq} ·qxns iylnz & ( 1) Êunnz unxzv énryééntztvn
xvvztv¡ntqn
& ( 1) = & ( 1) + & ( 1 − ) + + & ( 1 − )
Òv·tv zjr n
& ( 1) = & ( 1) + & ( 1 − ) + + & ( 1 − )
‰xtv ·zv & ( 1) = nxsq ≤ 1 ≤ q & ( 1) = k wévzqktvu xsy·jn Ívs•oy¹x• ëzquq xvvztv¡ntq¹uq ino zéylj
f·jxzsqk€n zévssnpiyxt€n iqsnz€
ojwvst¹nu |
xsnlyí•yí |
|
zjisq|y |
otj·ntqp |
& (N 1) |
ls¹ |
||||||||||
N = q 1 vz lv |
|
|
|
|
|
|
|
|
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
N = |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
N = |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
N = |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
N = |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
N = |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
N = |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Ìiéjzqu ktqujtqn tj tnrvzvéyí xquunzéqí ëzvp zjisq|€
Ívxrvs•ry wéq ojuntn rj lvp qo zén} |q{é tj nn lvwvstntqn lv xyuuj |q{é unt¹nzx¹ x 1 tj − 1 zv k€wvst¹nzx¹
éjkntxzkv & ( 1) = & ( − 1) ¬tjsvmq·tv k€wvst¹ízx¹ éjkntxzkj & ( 1) = & ( − 1)
ïzvi€ tjpzq znwné• ·qxsv x·jxzsqk€} iqsnzvk tjlv kvoknx zq k rkjléjz ·qxsj zénz•np xzévrq q xsv qz• wvsy·jí•qnx¹
énoys•zjz€ |
Æ xjuvu lnsn rj l€p x·jxzsqk€p iqsnz qunnz |
vlty q zy |
n xyuuy |q{é xzv¹•q} tj wnék€} zén} q wvxsnltq} |
zén} unxzj} Íyxz• ëzj xyuuj éjktj 1 ïqxsv xzv¹•nn tj 1 u unxzn zénz•np xzévrq tj¡np zjisq|€ wvrjo€kjnz xrvs•rv zén}otj·t€} ·qxns qunnz xyuuy |q{é 1 Êt€uq xsvkjuq vtv wvrjo€kjnz xrvs•rquq xwvxvijuq uv tv k€iéjz• wnékyí wv svkqty ¡nxzqotj·tvmv tvunéj iqsnzj fzvs•rquq n xwvxvijuq uv tv k€iéjz• kzvéyí wvsvkqty tvunéj iqsnzj Òjr rjr ëzq k€ivé€ tn ojkqx¹z léym vz léymj zv wv wéjkqsy wévqoknlntq¹ ·qxsv x·jxzsqk€} tvunévk x xyuuvp |q{é tj ·nzt€} unxzj}
& ( 1) ¬ zvmlj wv wéjkqsy xyuu€ vi•nn ·qxsv ©x·jxzsqk€}ª tvunévk éjktv
( + + + + + + + + + + + + + + ) =
Ívxrvs•ry kxnmv ¡nxzqotj·t€} tvunévk uqssqvt x·jxzsq k€u ¹ks¹nzx¹ wéqunétv vlqt iqsnz qo
béymj¹ {véuj vzknzj wvsy·jnzx¹ xsnlyí•qu viéjovu djun tqu wvxsnltqn zéq |q{é€ x·jxzsqkvmv iqsnzj q} lvwvstntq¹uq lv Ëjwéquné kunxzv kvo•unu Òvmlj nxsq
xyuuj wnék€} zén} |q{é éjktj 1 zv xyuuj wvxsnltq} zén} xzjtnz − 1 q wvsy·qzx¹ iqsnz xyuuj |q{é rvzvévmv éjktj
|
-sjkj ,9 è¬dÈÊcËʉ |
|
|
|
tjwéquné + + + + + = Òjr rjr |q{é€ k |
iqsnzn wéqtqujíz otj·ntq¹ vz lv ·qxsv |q{é k rj lvu
tvunén éjktv j q} xyuuj éjktj zv ·qxsv x·jxzsqk€} iqsnzvk éjktv & ( ) Ævz zvs•rv tj}v lntqn ëzvmv ·qxsj
lvkvs•tv zéylvnurv nxsq tn wvs•ovkjz•x¹ rvuw•íznévu ² ls¹ ëzvmv tjlv wvxzévnttyí tjuq zjisq|y otj·ntqp & (N 1) lv
xzévqz• lv otj·ntqp N =
Æ w u€ yrj nu ivsnn wévxzyí {véuy vzknzj ëzvp ojlj·q
Ënrvzvé€n xkvpxzkj ·qxns 2 3
ïqxsj &P(Q N) visjljíz é¹lvu xkvpxzk tjwvuqtjí•q} xkvpxzkj ·qxns &NQ êzv tn ylqkqzns•tv wvxrvs•ry k xqsy
wvxzévntq¹ jéq{unzq·nxrvmv zénymvs•tqrj qunnu & (Q N) = &N |
|||
|
|
|
Q |
Æ wnékyí v·nénl• vzunzqu ·zv &P(Q 1) vzsq·tv vz tys¹ |
|||
sq¡• wéq ≤ 1 ≤ Q(P − ) |
êzv xéjoy xsnlynz |
qo zvmv |
·zv |
rj lj¹ xsnlyí•j¹ xzévrj |
P jéq{unzq·nxrvmv |
zénymvs•tqrj |
|
lsqttnn wénl€ly•np tj P − |
|
|
|
bjsnn wvrj nu ·zv ·qxsj &P(Q 1) visjljíz xsnlyí•qu |
|||
xkvpxzkvu xquunzéqq |
|
|
|
&P(Q 1) = &P(Q Q(P − ) − 1) |
|
|
|
bs¹ ëzvmv xvwvxzjkqu rj lvuy Q otj·tvuy ·qxsy k P q·tvp
xqxznun x·qxsntq¹ ©lvwvstqzns•tvn ·qxsvª wvsy·jí•nnx¹ wy znu ojunt€ rj lvp |q{é€ nn lvwvstntqnu lv P − Ëjwéquné
k xnunéq·tvp xqxznun x·qxsntq¹ lvwvstqzns•t€u r
iylnz ·qxsv ‰xtv ·zv nxsq xyuuj |q{é ljttvmv
Q otj·tvmv ·qxsj éjkt¹sjx• 1 zv xyuuj |q{é lvwvstqzns•tvmv ·qxsj éjktj Q(P − ) − 1 Ívëzvuy Q otj·t€} ·qxns x xyuuvp
|q{é 1 xzvs•rv n xrvs•rv q x xyuuvp |q{é Q(P − ) − 1
Ëv ëzv q k€éj ntv éjkntxzkvu
Òjr rjr vi•nn rvsq·nxzkv Q otj·t€} ·qxns k P q·tvp
xqxznun x·qxsntq¹ éjktv PQ xu w zv qunnz unxzv xvvztv ¡ntqn
&P(Q ) + &P(Q ) + + &P(Q Q(P − )) = PQ |
|
bvrj nu znwné• xvvztv¡ntqn |
|
&P(N )&P(Q − N 1) + &P(N )&P(Q − N 1 − ) + |
|
+ &P(N 1)&P(Q − N ) = &P(Q 1) |
|
Ënrvzvé€n xkvpxzkj ·qxns 2 3 |
|
mln ≤ N ≤ Q bs¹ ëzvmv éjovi•nu kxn Q otj·t€n ·qxsj x xyuuvp
|q{é éjktvp 1 tj rsjxx€ Ç V uy rsjxxy vztnxnu ·qxsj xyuuj
wnék€} N |q{é rvzvé€} éjktj V Òvmlj xyuuj wvxsnltq} Q − N |q{é iylnz éjktj 1 − V Ív wéjkqsy wévqoknlntq¹ wvsy
·jnu ·zv k V p rsjxx k}vlqz &P(N V)&P(Q − N 1 − V) ·qxns Òjr
rjr vi•nn rvsq·nxzkv Q otj·t€} ·qxns x xyuuvp |q{é 1 éjktv &P(Q 1) zv wv wéjkqsy xyuu€ wvsy·jnu xvvztv¡ntqn Æ
·jxztvxzq wéq N = xvvztv¡ntqn wéqkvlqz r éjkntxzky
zjr rjr &P( 1) = wéq ≤ 1 ≤ P − q &P( 1) = wéq 1 < qsq 1 ≥ P
Ëjrvtn| wvrj nu ·zv qunnz unxzv éjkntxzkv
& (Q 1) = & & |
(Q 1 − Q) + & & |
P− |
(Q − 1 − Q + ) + + |
||||
P |
Q |
P− |
Q |
|
|
|
|
|
+ &V & |
(Q − V 1 − Q + V) + + &Q & |
P− |
( 1) |
|||
|
|
Q |
P− |
|
3 |
|
|
bs¹ ëzvmv éjovi•nu kxn Q otj·t€n ·qxsj k P q·tvp xqxznun
x·qxsntq¹ xyuuj |q{é rvzvé€} éjktj 1 tj rsjxx€ Ç V uy rsjxxy ≤ V ≤ Q vztnxnu ·qxsj k P q·tvp ojwqxq rvzvé€}
kxzén·jnzx¹ évktv V tysnp
Ëjplnu xrvs•rv ·qxns k}vlqz k V p rsjxx Çj lvn ·qxsv V mv rsjxxj uv nz i€z• k€iéjtv k lkj ëzjwj ftj·jsj k€inénu unxzj tj rvzvé€} xzv¹z tysq Òjr rjr éjxxujzéqkjízx¹ Q otj·t€n ·qxsj j rvsq·nxzkv tysnp éjktv V zv ëzv uv nz
i€z• xlnsjtv &VQ xwvxvijuq Ívxsn ëzvmv k€·nértnu kxn tysq q
yunt•¡qu rj |
lyí vxzjk¡yíx¹ |q{éy tj ΀ wvsy·qu |
(Q − V) otj·tvn |
·qxsv ojwqxjttvn |q{éjuq P − z n |
·qxsv k (P − ) q·tvp xqxznun x·qxsntq¹ xyuuj |q{é ëzvmv
·qxsj |
éjktj |
1 − (Q − V) = 1 − Q + V Çvsq·nxzkv |
zjrq} |
·qxns |
éjktv |
&P− (Q − V 1 − Q + V) Êo wévknlntt€} |
éjxxy |
lntqp |
|
|
|
V |
|
·qxns |
kqltv |
·zv k |
V p rsjxx k}vlqz &Q &P− (Q − V 1 − Q + V) |
||
Òjr rjr vi•nn rvsq·nxzkv Q otj·t€} ·qxns quní•q} xyuuy |q{é 1 éjktv &P(Q 1) zv k xqsy wéjkqsj xyuu€ wvsy·jnu
xvvztv¡ntqn |
|
|
|
||
Òjr rjr & |
|
(Q N) = &N zv qo xvvztv¡ntq¹ xsnlynz ·zv |
|||
|
|
Q |
|
|
|
& (Q |
N) = & &N−Q |
+ & &N−3+ + + &Q &N |
|||
|
|
Q Q |
Q Q− |
Q |
|
Æ€éj ntqn &P(Q N) ·néno iqtvuqjs•t€n rvë{{q|qntz€ ls¹ P = q z l wvsy·jnu wéqunt¹¹ {véuysy tnxrvs•rv éjo