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2.
r : n 1 a 1, a 2,..., a n1; n 2 b 1, b 2,..., b n2; n r r - x 1, x 2,..., x nr. n 1 n 2 n r (a j1, a j2,..., a jr), .
. . , . 10 , 2210=40 .
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. 2 n 1= n 2=...= n r= n, , U(n,r) = n r.
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. "" 28=256 .
2 n1 = n, n2 = n - 1, n3 = n - 2,..., nr = n r +1. , n(n - l)(n - 2) (n r +1). , :
, r n, n = r. () . , n a1,a2,...,an (n)n = n(n-1)... 21 .
, . n (n)n. (n)n (n)n = n(n-1)... 21 = n!
n! n -. 0! = 1.
.
1! = 1,
2! = 21 = 2,
3! = 321 = 6,
4! = 4321 = 24.
. . r r . , , . , , N=365. r . , r N , , , Nr. r , r . , r N (N)r. r = 23 (N)r = (365)23 = 365 364 363 343 = 36523.
. , , , , . , r = 4 (2,3,5,1), (2,3,5,4), (2,3,1,5), (2,3,5,1).
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12 13 14 15
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45.
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r n . , , n =2, r =3, n r = 23 = 8 :
111, 112, 121, 211, 222, 221, 212, 122.
:
{111}, {222}, {112, 121, 211}, {122, 212, 221}.
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n r n r r n () :
(r1, r2,..., rn), r1 + r2 +...+ rn = r. , n, r1, r 2 ,..., rn.
r(r1, r 2 ,..., rn).
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. , 12 (4 , 4 , 2 , 2 ) 12(4,4,2,2) = 207900.
, n r . r1, r 2 ,..., rn r1 + r2 +...+ rn = r 0 < r i, .
: x1 + x2 +...+ xn = r.
n r r n .
, r n .
n = 2, r = 3, , 23 = 8 , (.. ) . , , ; :
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111, 112, 122, 222 4 di
123, 124, 134, 234.
,
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r1 + r2 +...+ rn = r . :
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x1 + x2 +...+ xn = r
r n r n .
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r = 40, n = 3.
.
r n .
r n r1, r 2 ,..., rn , r1 + r2 +...+ rn = r, r k, 1≤ ≤ n, , . , r , . , r n r1, r 2 ,..., rn , r1 + r2 +...+ rn = r, r 1 r , r 2 r - r 1 r n r - r 1 - r 2 -...- r n-1 .
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, r n n r, r1, r 2 ,..., rn , r1 + r2 +...+ rn = r, 0≤ r k, , :
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.
, :
,
r1, r 2 ,..., rn , r1 + r2 +...+ rn = r, 0≤ r k, .
. n =2, r =3, n r = 23 = 8 ,
{111}, {222}, {112, 121, 211}, {122, 212, 221}. :
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.
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n .
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