Maths

Denote A as husband and B as wife and denote the numbers for the 5 pairs.

The ten people are:
A1, B1, A2, B2, A3, B3, A4, B4, A5, B5


(1) To select 4 persons in which there is no couple.

That means, the 4 persons are from different pairs.
Select 4 pairs out of 5 (1, 2, 3, 4, 5), there are ₅C₄ ways.
In each pair, there are 2 ways to select A or B.

The number of ways is ₅C₄ × 2⁴ = 5 × 16 = 80


(2) To select 4 persons in which there is exactly one couple.

First select the couple (1, 2, 3, 4, 5) in ₅C₁ ways.
Then, for the remaining 4 - 2 = 2 persons, they need to be from different pairs, with ₄C₂ ways.
In each pair, there are 2 ways to select A or B.

The number of ways is ₅C₁ × ₄C₂ × 2² = 5 × 6 × 4 = 120


(3) To select 4 persons in which there is exactly two couples.

Select the two couples (1, 2, 3, 4, 5) in ₅C₂ ways.
Then, select all four persons from the two couple.

The number of ways is ₅C₂ = 10


Notice that the numbers of ways in (1) and (2) and (3) add to be:
80 + 120 + 10 = 210

This is exactly the number of ways to select four persons from ten persons, that is, ₁₀C₄ = 210


2014-05-31 17:57:20 補充：
有興趣，但無時間，我要走了～

個問題係唔係New Progress in Senior Mathematics嘅英文數學書入面嘅第19課呢??

知足常樂 ,你用了 ncr!!!

2014-05-31 17:53:28 補充：
我正好有一題數學題想問你,不知你有沒有興趣解答

(1) No. of ways such that none of the 4 persons selected are husband and wife.
80
(2) No. of ways such that there is exactly 1 pair of husband and wife included.
160