() . .
= lim (m/n)
n→∞
, .
, , . m n E = {e1, e2,..., en}.
, . : ( , m , , ). . , , ( ) (.. ), . m n .
. ,
m , m- E, . ( ) n m, N(W) :
Cmn = n!/[m!(n - m)!] = n(n - 1)...(n - m + 1)/m!.
Cmn, , , :
Cmn = Cn-mn ( ),
Ckn+1 = Ckn + Ck-1n; C0n = 1 ( ),
C0n + C1n +... + Cnn = 2n ( ).
1. 10 . ? , a?
( 10 3):
N(W) = C310 = 10×9×8/(1×2×3) = 120.
A - , a. ( a), .. 9 2: N(A) = C29 = 9×8/2 = 36.
|
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,
(A) = N(A)/N(W) = 36/120 = 0,3.
. ,
m , , m- , , . ( ) n m, N(W) :
Amn = Cmn×m! = n!/(n - m)! = n(n - 1)...(n - m + 1).
n = m, , .. . N(W) = Ann = n!.
2. , 8 , . , ?
. 8 , N(W) = A88 = 40320. , : 8 , , 6 , N(A) = 2×8×6!. (A) = N(A)/N(W) = 2/7.
. ,
m E = {e1, e2,..., en}, , m- , . . , m = 4 {e1, e1, e2, e1} {e2, e1, e1, e1} , {e1, e1, e3, e1} . , N(W) = Cmn+m-1.
3. 16 . . , , : - , - .
. , , 16 4, .. N(W)= C416+4-1 = C419.
, A, 16, (A) = N(A)/N(W) = C416/C419 0,47.
, , 16, (A) = N(A)/N(W) = C116/C419 0,004.
. ,
m E = {e1, e2,..., en}, , m- ( , ), , . , m = 4 {e1, e1, e2, e1}, {e2, e1, e1, e1} {e1, e1, e3, e1} . , ,
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N(W)= nm.
4. E = {, , , , } . , ?
. , 5 4 .. N(W)= 54. . (A) = N(A)/N(W) = 1/54 0,0016.
.
E m . , E s E1, E2,..., Es , :
. i ni , i = 1, 2,..., s.
. i ni.
. i, , . , n = 7, n1 = 2, n2 = 2, n3 = 3 {E1 = {e1, 2}, 2 = {e3, 4}, 3 = {e5, 6, e7}} {E1 ={e3, 4}, 2 ={e1, 2}, 3 = {e5, 6, e7}} .
N(W) = n!/(n1! × n2! ×... × ns!).
5. , , . ? , ?
. : s = 3, n = 10, n1 = 3, n2 = 3, n3 = 4. N(W) = 10!/(3!×3!×4!) = 4200.
- . : s = 3, n = 8, n1 = 3, n2 = 3, n3 = 2. N(A) = 8!/(3!×3!×2!) = 560. (A) = N(A)/N(W) = 560/4200 = 2/15.