I28-53

Question

A certain company is forming a committee of  n  people to be chosen from  n  married couples, where  n  is a two-digit even number.
 
Select for  No married couples  the number of different committees that can be formed if no two people who are married to each other are allowed to serve on the committee, and select for  Only married couples  the number of different committees that can be formed if the committee must consist only of married couples. Make only two selections, one in each column.

Bunuel · Answer

Official Solution:

Since the committee must not include any married couples, only one "representative" from each married couple can be selected. We are forming an  n -person committee, so we must select  n  married couples that will each send one "representative" to the committee. The number of ways to choose  n  couples out of  n  is  C(n, n) = 1 .
 
For each of these  n  couples, there are two choices for the "representative": either husband or wife. This results in 2 * 2 * ... * 2 ( n  times)  = 2^n  possibilities.
 
Therefore, the total number of ways to form the committee is: 
   
 C(n, n) * 2^n = 2^n .
   
As for the committees that can be formed if the committee must consist only of married couples, we are choosing  \frac{n}{2}  couples (which gives us  n  people) from  n  married couples. This is given by:
   
 C(n, \frac{n}{2}) = 
 
 = \frac{n!}{\frac{n}{2}! * (n - \frac{ n}{2})!} = 
 
 = \frac{n!}{(\frac{n}{2}!)^2}

Correct answer:

No married couples  " 2^n "

Only married couples  " \frac{n!}{(\frac{n}{2}!)^2} "

ArmaanBajaj · Answer

I don’t quite agree with the solution. Logic not fully clear

No married couples	Only married couples
		\(n!\)
		\(n^2\)
		\(2^n\)
		\(\frac{n!}{2(n-2)!}\)
		\(\frac{n!}{(\frac{n}{2}!)^2}\)
		\(\frac{(2n)!}{(n!)^2}\)

No married couples	Only married couples
		\(n!\)
		\(n^2\)
		\(2^n\)
		\(\frac{n!}{2(n-2)!}\)
		\(\frac{n!}{(\frac{n}{2}!)^2}\)
		\(\frac{(2n)!}{(n!)^2}\)

I28-53

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