Question

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

• A. $\frac{(200!)}{2^{100}\left(100\right)!}$
• B. $\mathrm{1.3.5}\dots \dots 199$
• C. $\frac{(200!)}{\left(100\right)!}$
• D. $\left(\frac{101}{2}\right),\left(\frac{102}{2}\right)\dots \dots \left(\frac{200}{2}\right)$

Answer 1

Answer: (A), (D)

The correct options are

A $\frac{(200!)}{2^{100}\left(100\right)!}$

B. $\mathrm{1.3.5}\dots \dots 199$

D $\left(\frac{101}{2}\right),\left(\frac{102}{2}\right)\dots \dots \left(\frac{200}{2}\right)$

Firstly, number of ways in which $200$ people can be arranged at $200$ places is ${}^{200}{P}_{200}$

$=200!\dots \dots \left(1\right)$

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

${=}^{100}{P}_{100}$

$=100!\dots \dots \left(2\right)$

We have to divide (1) by (2)

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,

$=[2\times 2\times 2\times \dots \dots ]100$ times

$=2^{100}\dots \dots \left(3\right)$

Thus, total number of ways in which $200$ people can be arranged in $100$ couples is

$=\frac{(200!)}{2^{100}\left(100\right)!}$

Therefore, required number of ways $=\frac{(200!)}{2^{100}\left(100\right)!}$
$=\frac{\left(100\right)!\left(101\right)\left(102\right)\left(103\right).......\left(200\right)}{2^{100}(100!)}$

$=\left(\frac{101}{2}\right)\left(\frac{102}{2}\right)\dots \dots \left(\frac{200}{2}\right)$
So, Option $D$ also a correct answer.

And also$\frac{(200!)}{2^{100}\left(100\right)!}$ is in the form of $\frac{\left(2n\right)!}{2^n(n!)}$, Here $n=100$
As we know that $\frac{\left(2n\right)!}{2^n(n!)}=1\cdot 3\cdot 5\cdot 7\cdot \cdot \cdot \cdot \cdot \cdot (2n-1)$
Since $n=100$
So, $\frac{(200!)}{2^{100}\left(100\right)!}=1\cdot 3\cdot 5\cdot 7\cdot \cdot \cdot \cdot \cdot \cdot 199$
Therefore, options $A,B,D$ are correct answers.