First slide

Binomial theorem for positive integral Index

Question

The coefficient of $x^{7}$ in the expansion of ${(1 + \frac{1}{1!} x + \frac{1}{2!} x^{2} + \frac{1}{3!} x^{3} + \frac{1}{4!} x^{4} + \frac{1}{5!} x^{5})}^{2}$ is

Moderate

A

$\frac{2}{7!} (^{8} C_{3})$
B

$\frac{2^{10}}{5!}$
C

$\frac{4}{15}$
D

$\frac{7}{20!}$

Solution

Required coefficient is
$\begin{aligned} \frac{1}{2! 5!} + \frac{1}{3! 4!} + \frac{1}{4! 3!} + \frac{1}{5! 2!} \\ = \frac{1}{7!} (^{7} C_{2} +^{7} C_{3} +^{7} C_{4} +^{7} C_{5}) \\ = \frac{2}{7!} (^{7} C_{2} +^{7} C_{3}) = \frac{2}{7!} (^{8} C_{3}) \end{aligned}$

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