Permutation and Combination

a)The 1st one can be any people , 10 waysThe 2nd one have 8 choices since the 1st one couple cann't be choose.The 3rd one have 6 choices since the 1st and the 2nd two couples cann't be choose.The 4th one have 4 choices since 3 couples cann't be choose.4 peoples have 4! permutation.10 x 8 x 6 x 4 / 4! = 80 ways.b)One couple have 5 choices.Then choose 2 peoples among 4 couples but they cann't be couple ,8C2 - 4 = 24 waysor8 x 6 / 2 = 24 ways;The number of ways that has exactly one couple :5 x 24 = 96 ways


2010-11-01 22:54:15 補充：
Corrections :

5 x 24 should be 120 ways

Sorry!!

2010-11-03 03:29:18 補充：
(a) 為什麼要除 4! permutation?

因為4個人有4!個排列,10 x 8 x 6 x 4 包含了4人的全部排列,
如 1234 , 4321 , 1342 , 4213 .... 其實算同一種。重複了 4! = 24次。

(b) 為什麼減4而不是減2?

所選二人組有4個情況不行,就是那 4 對couple。

a club, there are 5 couples. It is to make a committee that consists of 4 people.

(a) Find the number of ways that has no couples.
(b) Find the number of ways that has exactly one couple