Question

There are 3 men and 7 women taking a dance class. If N is the 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, then the value of $\dfrac{N}{90}$ is.

Answer 1

Hint: We will start by first finding the ways in which we can select 3 women’s out of 7. Then we will distribute the 3 men’s with them and finally we will find ways of grouping 4 women in pairs of 2.

Complete step-by-step answer:
Now, it has been given to us that there are 3 men and 7 women taking a dance class. So, we know that ways of selecting 3 women’s out of 7 women’s in \[{}^{7}{{C}_{3}}\] as the ways of selecting r objects out of n object is \[{}^{n}{{C}_{r}}\]. Now, the ways of permuting 3 men’s is 3!.
Also, it has been given that the rest 4 women are paired up with each other. This means that we have to find the number of ways in which 4 women can be paired in (2, 2). So, we have $\dfrac{4!}{2!\times 2!}$ but since the groups are identical i.e. it doesn’t matter which two women we select first, only matters is which women we select not the order. So, we have the cases now reduced by 2! as $\dfrac{4!}{2!\times 2!\times 2!}$.
Now, by using the multiplication principle we can obtain the total ways of pairs 3 men and 7 women and the remaining two as pairs. So, we have,
${}^{7}{{C}_{3}}\times 3!\times \dfrac{4!}{2!\times 2!\times 2!}$
Now, we know that,
$\begin{align}
  & {}^{n}{{C}_{r}}=\dfrac{n!}{\left( n-r \right)!\times r!} \\
 & \Rightarrow \dfrac{7!}{3!\times 4!}\times 3!\times \dfrac{4!}{2!\times 2!\times 2!} \\
 & \Rightarrow \dfrac{7!}{2!\times 2!\times 2!} \\
 & \Rightarrow \dfrac{3\times 4\times 5\times 6\times 7}{2\times 2} \\
 & \Rightarrow 3\times 5\times 6\times 7 \\
 & \Rightarrow 21\times 30 \\
 & \Rightarrow 630 \\
\end{align}$
Therefore, the value of N is 630.
$\begin{align}
  & \Rightarrow \dfrac{N}{90}=\dfrac{630}{90} \\
 & \Rightarrow \dfrac{N}{90}=7 \\
\end{align}$

Note: It is important to note that we have used the multiplication principle to find the total ways by finding the ways for each sub case and multiplied each of them to find the total ways.