If the second term in the expansion ${(a^{1 / 13} + a^{3 / 2})}^{n}$ is $14 a^{5 / 2}$ , then the value of $\frac{{^{n} C}_{3}}{{^{n} C}_{2}}$ is .........

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Solution

Second Term of the expansion is
$T_{2} =^{n} C_{1} \times {(a)}^{\frac{1}{13}} \times {(a)}^{\frac{3 (n - 1)}{2}} = 14 a^{\frac{5}{2}}$
So We can compare both Side so
$^{n} C_{1} = 14$ So $n = 14$
To find $\frac{^{14} C_{3}}{^{14} C_{2}} = \frac{14 - 3 + 1}{3} = 4$ [Note :-Important Formula $\frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{(n - r + 1)}{r}]$
So Answer is $4.$

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