Solveeit Logo
Login

Question

Question: The maximum number of permutations of 2n letters in which there are only a's and b's, taken all at a...

The maximum number of permutations of 2n letters in which there are only a's and b's, taken all at a time given by :

A

2nCn{}^{2n}C_n

B

21⋅62⋅103...4n−6n−1⋅4n−2n\frac{2}{1}\cdot\frac{6}{2}\cdot\frac{10}{3}...\frac{4n-6}{n-1}\cdot\frac{4n-2}{n}

C

n+11⋅n+22⋅n+33⋅n+44...2n−1n−1⋅2nn\frac{n+1}{1}\cdot\frac{n+2}{2}\cdot\frac{n+3}{3}\cdot\frac{n+4}{4}...\frac{2n-1}{n-1}\cdot\frac{2n}{n}

D

2n.[1.3.5..........(2n−3)(2n−1)]n!\frac{2^n.[1.3.5..........(2n-3)(2n-1)]}{n!}

Answer

(2nn)\displaystyle \binom{2n}{n}

Explanation

Solution

If we want as many distinct arrangements as possible, we should use the same number of a’s and b’s (because if the counts are very different, the number of arrangements given by the multinomial coefficient is lower). Thus, we take n a’s and n b’s. The number of distinct arrangements is

(2nn),\binom{2n}{n},

which is exactly option (A).