Combinations of Married Couples

Date: 11/08/96 at 02:33:40
From: Jingde Chu
Subject: Combinations of couples

Given that there are 6 married couples:   

a. What is the probability that the 12 people can be grouped 
   into 6 pairs where each pair is a married couple?

b. If we select only 4 people out of the 12, what is the 
   probability that:

      i. none of them are married couples?
     ii. there is only one pair of married couples?
    iii. there are two pairs of married couples?

Date: 11/08/96 at 08:23:29
From: Doctor Donald
Subject: Re: Combinations of couples

>a. What is the probability that the 12 people can be grouped 
    into 6 pairs where each pair is a married couple?

Suppose the men are denoted by a, b, c, d, e, f and the women are 
denoted by A, B, C, D, E, F where a is married to A and so on.

Starting with man a, there is 1/6 chance he is matched with his 
wife A.  Move to man b, the chance that he is matched with B from 
the remaining 5 women is 1/5.  Move to man c, the chance that he is 
matched with C is 1/4, and so on.    

The total probability that each pair is a married couple is:

   1/6 x 1/5 x 1/4 x 1/3 x 1/2 x 1/1 = 1/720.

>b. If we select only 4 people out of the 12, what is the 
    probability that:

    i. none of them are married couples:

Consider the couples   a  b  c  d  e  f
                       A  B  C  D  E  F

First we select 4 columns, then one item from each column.

The number of ways to select 4 from 6 columns is 6_C_4 = 15.
The number of ways to select one member from the pair is 2_C_1 = 2.

The number of ways to select 4 people, none of whom are a pair, is 
15 x 2 = 30.

The number of groups of 4 people chosen without restriction from 12 
people is  (1/3)12_C_4 = 165.

The probability that there are no married couples is 30/165 = 2/11.

   ii. there is only one pair of a married couples:

Now we must select one column from the six (in 6 ways) and then two 
other columns from the other 5. This can be done in 5_C_2 = 10 ways.  
Of these, there are two ways to select one member of the pair.  The 
total number of ways to select just one pair is 6 x 10 x 2 = 120.

The probability that there is only one pair of married couples is
120/165 = 8/11.

  iii.  there are two pairs of married couples:

We could get this from finding 1 - 2/11 - 8/11 = 1/11, but we can also 
calculate it directly.

We must select 2 columns from the six.  This can be done in 15 ways, 
and so the probability that there are two pairs of married couples is 
15/165 = 1/11.

-Doctor Donald,  The Math Forum
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