推 ooopppeeennn:謝啦~~ 03/22 22:00
※ 引述《ooopppeeennn (open)》之銘言：
: (a) Find the number of n-dight words generated from the alphabet {0,1,2}
: in each of which none of the dights appears exactly three times.
: (b) Find the number of n-dight words generated from the alphabet {0,1,2,3,4}
: in each of which the total number of 0's and 1's is even
(a). (e^x-x^3/3!)^n 求x^n/n! 之係數......
x -x x -x
(b) e +e 2 x x x e -e x x x
(----------) e e e + (---------)e e e
2 2
= 1/4(e^2x+2+e^-2x)e^xe^xe^x + 1/4(e^2x-2+e^-2x)e^xe^xe^x
= 1/2e^5x + 1/2e^x
∞ ∞
= 1/2 Σ 5x^n/n! + 1/2Σ x^n/n!
n=0 n=0
求 x^n/n! 之係數 = 1/2(5^n+1)
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