※ 引述《wisely13 (two-6)》之銘言：
: 1.Using the summation operator method to find sigma i^5 (for all i=1 to n)
: 2.Find the coefficient of x^24 in each of the following:
: (a) (1+x+x^2+x^3+......+x^6)^5
: (b) (1+2x)^2/(1-2x)^4
by 1/(1-2x) = 1 + 2x + (2x)^2 + (2x)^3 + ...
∞
= Σ 2^k * x^k
k=0
we have
∞ k+3
1/(1-2x)^4 = Σ ( ) * 2^k * x^k
k=0 k
so
(1+2x)^2/(1-2x)^4
∞ k+3
= (1+2x)(1+2x) * Σ ( ) * 2^k * x^k
k=0 k
thus , the coefficient of x^24 is
27 26 25
( ) * 2^24 + 2* ( ) *2^23 + 4 * ( ) * 2^22
24 23 22
: (c) (x^3+x)/(1-2x)^3
∞ k+2
(x^3+x)/(1-2x)^3 = (x^3+x) Σ ( ) * 2^k * x^k
k=0 k
the coefficient of x^24 is
23 25
( ) * 2^21 + ( ) * 2^23
21 23
: (d) 1/(1+x)(1-3x)(2x+3)
let 1/(1+x)(1-3x)(2x+3) = c1/(1+x) + c2/(1-3x) + c3/(2x+3)
we have c1 = 1/4 , c2= 9/44 , c3= -4/11
∞
thus , 1/(1+x)(1-3x)(2x+3) = 1/4 * Σ (-1)^k * x^k
k=0
∞
+ 9/44 * Σ 3^k * x^k
k=0
∞
- 4/11 * 1/2 * Σ (-3/2)^k * x^k
k=0
so the coefficient of x^24 is
1/4 + 9/44 * 3^24 - 4/22 * (-3/2)^24
: (e) (1-2x)^(-4/3)
∞ α
by (1+z)^α = Σ ( ) z^k
k=0 k
∞ -4/3
so we have (1-2x)^(-4/3) = Σ ( ) (-2x)^k
k=0 k
thus the coefficient of x^24 is
-4/3 -4/3 (-4/3)(-4/3 -1) ...(-4/3 -24 +1)
( ) * (-2)^24 , where ( ) = ---------------------------------
24 24 24!
: 3.Prove that the number of partitions of r into parts each of which appears
: at most twice is equal to the number of partitions of r into parts the
: sizes of which are not divisible by 3
: 4.How many 20-digit ternary (0,1,2) sequences are there where:
: (a) There is at least on 2 and odd number of 0's?
: (b) No symbol occurs exactly twice?
: (c) No symbol occurs exactly three times?
: (d) There are exactly two 2's or none at all?
: 5.Suppose that X is a discrete random variable with probability
: distribution given by Pr(X=x)={ k(1/4)^x, x=0,1,2.....}
: 0, otherwise
: where k is a constant, determine
: (a) k=? (b)母體平均? 標準差?
: 6.What probability can we select six nonconsecutive integers from {1,2,3,..,37}
: 7.If Y is geometric random variable with E(Y)=7/3, determine
: (a) Pr(Y=3)
: (b) Pr(Y>=3)
: (c) Pr(Y>=5)
: (d) Pr(Y>=5 | Y>=3)
: (e) Pr(Y>=6 | Y>=4)
: (f) Var(Y)
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