First slide

Binomial theorem for positive integral Index

Question

The coefficient of $x^{20}$ in the expansion of ${(1 + \frac{1}{1!} x + \frac{1}{2!} x^{2} + \dots + \frac{1}{20!} x^{20})}^{3}$ is

Moderate

A

0
B

$\frac{3}{20!}$
C

$\frac{3^{20}}{20!}$
D

$^{60} C_{20}$

Solution

Note that, coefficient of $x^{20}$ in
${(1 + \frac{1}{1!} x + \frac{1}{2!} x^{2} + \dots + \frac{1}{20!} x^{20})}^{3}$
= coefficient of $x^{20}$ in
${(1 + \frac{1}{1!} x + \frac{1}{2!} x^{2} + \dots + \frac{1}{20!} x^{20} + \frac{1}{21!} x^{21} + \dots)}^{3}$
= coefficient of $x^{20}$ in $e^{3 x} =$ $\frac{3^{20}}{20!}$

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