Question

There are three men and seven women taking a dance class. Number of different ways in which each man is paired with a woman partner, and the four remaining women are paired into two pairs each of two is

• A. $105$
• B. $315$
• C. $630$
• D. $450$

Answer 1

The correct option is C

$630$

Evaluate the number of ways using Combinations

Given, $3$ women can be selected in ${}^7\mathrm{C}_3$ ways and can be paired with $3$ men in $3!$ ways.

Remaining $4$ women can be grouped into two couples as

$\frac{4!}{2!\times 2!\times 2!}=3$

Therefore, total number of ways

$$\begin{gathered} ={}^7\mathrm{C}_3\times 3!\times 3 \\ =\frac{7!}{(7-3)!\times 3!}\times 3!\times 3 \\ =\frac{7!}{4!}\times 3 \\ =5\times 6\times 7\times 3 \\ =630 \end{gathered}$$

Total number of ways$=630$

Hence option $\left(C\right)$, $630$ is the correct answer.