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 explanation for the correct answer.

Find the number of ways to form $15$ couples.

Given: Total men are $15$ and total women are $15$

Number of ways for the first couple $={}^{15}{\mathrm{C}}_1\times {}^{15}{\mathrm{C}}_1={15}^2\mathbf{(}\mathbf{\because }{}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{1}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{n}\mathbf{)}$

Number of ways for the second couple $={}^{14}{\mathrm{C}}_1\times {}^{14}{\mathrm{C}}_1$

Therefore a number of ways for $15$ couples $={15}^2+{14}^2+{13}^2+......+1=1240$

Hence option $\left(A\right)$ is the correct answer.