First slide

Binomial theorem for positive integral Index

Question

The coefficient of $x^{20}$ in ${(1 + 3 x + 3 x^{2} + x^{3})}^{20}$ is

Moderate

A

$^{60} C_{40}$
B

$^{30} C_{20}$
C

$^{15} C_{2}$
D

none of these

Solution

We have ,
${(1 + 3 x + 3 x^{2} + x^{3})}^{20} = {\{(1 + x)^{3}\}}^{20} = (1 + x)^{60}$
$∴$ Coefficient of $x^{20}$ in ${(1 + 3 x + 3 x^{2} + x^{3})}^{20}$
=Coefficient of $x^{20}$ in $(1 + x)^{60} =^{60} C_{20} =^{60} C_{40}$

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