Find the ratio $^{20} C_{r} t o^{25} C_{r}$ when each of them has the greatest value possible.
If the ans is $a / b$ , find $a + b$

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Solution

Greatest value of $^{20} C_{r} =^{20} C_{10}$
and greatest value of $^{25} C_{r} =^{25} C_{12} =^{25} C_{13}$
$∴ \frac{^{20} C_{10}}{^{25} C_{12}} = \frac{13! 12! 20!}{10! 10! 25!} = \frac{11 \times 13}{7 \times 23 \times 25} = \frac{143}{4025}$
$\Rightarrow a + b = 4168$

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