Question

What is the ratio of ${}^{20}{C}_r$ to ${}^{25}{C}_r$ when each of them has the greatest value possible?

• A. $\frac{143}{4000}$
• B. $\frac{143}{4025}$
• C. $\frac{4}{5}$
• D. none of these

Answer 1

Answer: (B)

$\frac{143}{4025}$

$n⟶$Even

${}^n{C}_r$ is maximum when $r=\frac{n}{2}$

$n⟶$ odd

${}^n{C}_r$ is maximum when $r=\frac{n+1}{2}$ or $\frac{n-1}{2}$

Maximum value of $\left[\frac{{}^{20}{C}_r}{{}^{25}{C}_r}\right]$

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

$=\frac{\frac{20!}{10!\times 10!}}{\frac{25!}{13!\times 12!}}=\frac{13\times 11\times 12\times 12\times 11}{25\times 24\times 23\times 22\times 21}=\frac{143}{4025}$