Question

Evaluate $\frac{{}^{20}{C}_r}{{}^{25}{C}_R}$ when both numerator and denominator have their greatest values.

Answer 1

$\frac{{}^{20}{C}_r}{{}^{25}{C}_r}$

Greatest value of ${}^{20}{C}_r$ is at $r=10$.

Greatest value of ${}^{25}{C}_r$ is at $r=12$.

$\frac{{}^{20}{C}_{10}}{{}^{25}{C}_{12}}=\frac{\frac{20!}{10!\times 10!}}{\frac{25!}{12!\times 13!}}$

$=\frac{20!\times 12!\times 13!}{10!\times 10!\times 25!}$

$=\frac{12\times 11\times 13\times 12\times 11}{25\times 24\times 23\times 22\times 21}$

$=\frac{13\times 11}{25\times 23\times 7}$

Greatest value$=\frac{143}{4025}$