Question

Number of ways in which $200$ people can be divided in $100$ couples is

• A. $\frac{\left(200\right)!}{2^{100}\left(100\right)!}$
• B. $1\times 3\times 5....199$
• C. $\left(\frac{101}{2}\right)\left(\frac{102}{2}\right)....\left(\frac{200}{2}\right)$
• D. $\frac{\left(200\right)!}{\left(100\right)!}$

Answer 1

The correct option is A $\frac{\left(200\right)!}{2^{100}\left(100\right)!}$

1) Firstly, number of ways in which 200 people can be arranged at 200 places is ${{200}_P}_{200}$

$=\frac{200!}{(200-200)!}$

$=\frac{200!}{0!}=200!$ (1)

2) Now, these 200 people have been divided into 100 couples and order of couples is immaterial.

Thus, number of ways in which these 100 couples can be arranged at 100 places is ${{100}_P}_{100}$

$=100!$ (2)

We have to divide (1) by (2)

3) Again, each pair of couple can be arranged in two different ways as order in which couple is formed is immaterial.

Thus, number of ways in which 100 couples can be arranged within themselves is, $2\times 2\times 2\times ---------100times$

$=2^{100}$ (3)

We have to divide (1) by (3)

Thus, total number of ways in which 200 people can be arranged in 100 couples is $\frac{200!}{2^{100}\times 100!}$