推 iclaire : 感謝你!!! 03/01 10:40
1.若有8顆相同的紅球，10顆相同的白球，5顆相同的黑球，求下列方法數：
(1) 至少取一球
C(m,n) = m!/m!(m-n)!
二項式: (x + 1)^n = C(n,0) + C(n,1)x + C(n,2)x^2 + ...
(1 + 1)^23 = C(23,0) + C(23,1) + C(23,2) + ... + C(23,23)
至少取一球 = C(23,1) + C(23,2) + ... + C(23,23)
= 2^23 - C(23,0)
= 2^23 - 1
(2) 每種顏色至少一
Red至少一 = (1 + 1)^8 - C(8,0) = 2^8 - 1 = 255
White至少一 = (1 + 1)^10 - C(10,0) = 2^10 - 1 = 1023
Black至少一 = (1 + 1)^5 - C(5,0) = 2^5 - 1 = 31
ans = 255*1023*31
(3) 甲乙兩人分球(要分完)且每人至少一顆
(1 + 1)^23 = C(23,0) + C(23,1) + C(23,2) + ... + C(23,22) + C(23,23)
a = 每人至少一顆
= C(23,1) + C(23,2) + ... + C(23,22)
= 2^23 - C(23,0) - C(23,23)
= 2^23 - 1 - 1
= 2^23 - 2
b = a / 重疊
= (2^23 - 2) / 8 * 10 * 5
2.
B = { (x+2y,x-y)| x-2y=-2 } => x = 2y - 2
B = (u,v)
u = x + 2y
= (2y - 2) + 2y
= 4y - 2
v = x - y
= (2y - 2) - 2y
= -2
A = { (x,y)| x-y=9 }
=> x = y + 9
= -2 + 9
= 7
ans = (7, -2)
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