First slide

Binomial theorem for positive integral Index

Question

The coefficient of $x^{n}$ $i n t h e e x p a n s i o n o f$ ${(1 + \frac{1}{1!} x + \frac{1}{2!} x^{2} \dots + \frac{1}{n!} x^{n})}^{2}$

Moderate

A

$\frac{2^{n}}{n!}$
B

$\frac{2^{n}}{n}$
C

$n!$
D

$\frac{1}{n!}$

Solution

Coefficient of $x^{n}$ in
${(1 + \frac{1}{1!} x + \frac{1}{2!} x^{2} + \dots + \frac{1}{n!} x^{n})}^{2}$
= coefficient of $x^{n}$ is
${(1 + \frac{1}{1!} x + \frac{1}{2!} x^{2} + \dots + \frac{1}{n!} x^{n} + \frac{1}{(n + 1)!} x^{n + 1} \dots)}^{2}$
= coefficient of $x^{n}$ in = $e^{2 x} = (1 + 2 x + \frac{2^{2} x^{2}}{2!} + \frac{2^{3} x^{3}}{3!} \dots)$
$= \frac{2^{n}}{n!}$
Alternate Solution
Coefficient of $x^{n}$ in
$\begin{array}{l} (1 + \frac{1}{1!} x + \frac{1}{2!} x^{2} + \dots + \frac{1}{n!} x^{n}) \\ (1 + \frac{1}{1!} x + \frac{1}{2!} x^{2} + \dots + \frac{1}{n!} x^{n}) \\ = \frac{1}{n!} + \frac{1}{1! (n - 1)!} + \frac{1}{2! (n - 2)!} + \dots + \\ = \frac{1}{n!} [^{n} C_{0} +^{n} C_{1} +^{n} C_{2} + \dots^{n} C_{n - 1} +^{n} C_{n}] = \frac{2^{n}}{n!} \end{array}$

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