※ 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
將其展開 其中某一項為
5!
-------------- * 1^a * x^(b+2c+3d+4e+5f+6g) a+b+c+d+e+f+g = 5
a!b!c!d!e!f!g!
聯立解 a+b+c+d+e+f+g = 5
b+2c+3d+4e+5f+6g = 24
有解 (a,b,c,d,e,f,g) = (1,0,0,0,0,0,4) , (0,1,0,0,0,1,3)
(0,0,1,0,1,0,3) , (0,0,0,2,0,0,3)
故 x^24 的係數為
5! 5! 5! 5!
------ + ----- + ----- + ------
4! 3! 3! 2!3!
: (b) (1+2x)^2/(1-2x)^4
: (c) (x^3+x)/(1-2x)^3
: (d) 1/(1+x)(1-3x)(2x+3)
: (e) (1-2x)^(-4/3)
: 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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