Question

The ratio of ${}^{24}{C}_r{:}^{25}{C}_r$ when each of them has the greatest value possible is:

• A. 12:25
• B. 13 : 25
• C. 13 : 24
• D. 1 : 2

Answer 1

Answer: (B)

13 : 25

Note that ${}^{2n}{C}_r$ is maximum if and only if $r=n$ and ${}^{2n-1}{C}_r$ is maximum if and only if $r=n-1$ or $r=n$.
Hence ${}^{24}{C}_r$ is maximum when $r=12$ and ${}^{25}{C}_r$ is maximum when $r=12$.
The ratio ${}^{24}{C}_{12}:{}^{25}{C}_{12}$ is :
$\frac{{}^{24}{C}_{12}}{{}^{25}{C}_{12}}=\frac{\frac{24!}{12!12!}}{\frac{25!}{12!13!}}$

$=\frac{24!}{12!12!}\times \frac{12!13!}{25!}$
$=\frac{13}{25}$