Question

The number of ways of dividing $15$ men and $15$ women into $15$ couples, each consisting of a man and a woman, is

• A. $1240$
• B. $1840$
• C. $1820$
• D. $2005$

Answer 1

The correct option is A $1240$

The number of ways of choosing first couple is $\left({}^{15}{C}_1\right)\left({}^{15}{C}_1\right)={15}^2$

The number of ways of choosing second couple is $\left({}^{14}{C}_1\right)\left({}^{14}{C}_1\right)={14}^2$, and so on.

Thus, the number of ways of choosing the couples is

${15}^2+{14}^2+{13}^2+...+2^2+1^2=\frac{15\times (15+1)(2\left(15\right)+1)}{6}=1240$