# Selecting 5 pairs of men and 5 women from 10 women and 12 men

## Question

A dance class consists of $22$ students, of which $10$ are women and $12$ are men. If $5$ men and $5$ women are to be chosen and then paired off, how many results are possible?

## Approach

According to me, the number of results possible is:

$$\binom{10}{5}*\binom{12}{5}*5!*2^{5}$$

$$\binom{10}{5}*\binom{12}{5}*5!$$

## My conclusion

Shouldn't be there $2$ options in each pair i.e ordering between men and women for $5$ such group, making it $5!$? Why is the answer not leaving $5!$? Are they not considering order? And if the order is important, is my answer correct in this case?

• Why would the be two options in each pair . A pair is of combination of a man and a woman . Even if you select fist a man then a woman to pair him or select first a woman and then a man to pair her both are same – ATHARVA Jun 1 '17 at 13:04
• if order is important, ur answer is right i believe – Kiran Jun 1 '17 at 13:11
• Yes, your answer is correct if "Fred and Ginger" is counted as a distinct pairing from "Ginger and Fred". – Michael Seifert Jun 1 '17 at 13:49

$$\binom{10}{5}*\binom{12}{5}*5!*2^{5}$$ is the right answer if the order in which you pick the pair is important. For example, if(x,y) and (y,x) are different.
First select five men and women. Then select one man $M_1$ at random, and pair him off with one of the 5 different women. Next select another man $M_2$, and assign him to one of the 4 remaining women. Continuing this, there are $5!$ ways to pair off the 5 selected men and women. As such, the total number of pairs equals:
$${10 \choose 5}{12 \choose 5}5!$$