Question

The maximum number of permutations of $2n$ letters in which there are only $a^{\prime }s$ and $b^{\prime }s$. taken all at a time is given by :

• A. ${{}^{2n}C}_n$
• B. $\frac{2}{1}.\frac{6}{2}.\frac{10}{3}....\frac{4n-6}{n-1}.\frac{4n-2}{n}$
• C. $\frac{n+1}{1}.\frac{n+2}{2}.\frac{n+3}{3}\frac{n+4}{4}.....\frac{2n-1}{n-1}.\frac{2n}{n}$
• D. $\frac{2^n\left[\mathrm{1.3.5....}\right(2n-3\left)\right(2n-1\left)\right]}{n!}$

Answer 1

The correct option is A ${{}^{2n}C}_n$

Given $2n$ letters

Are in the form of $a$ & $b$.

Let us suppose that $x$ as we have

Then we will have $2n-x$ $bs$

i.e, $\frac{2n!}{x!(2n-x)!}{=}^{2n}{C}_x$ $[{\because }^n{C}_x=\frac{n!}{x!(n-x)!}]$

${}^{2n}{C}_x$ is maximum when $x$ is half of $2n$

i.e; $x=\frac{2n}{2}=n$

so we can write ${}^{2n}{C}_x{=}^{2n}{C}_n$