First slide

Binomial theorem for positive integral Index

Question

$^{404} C_{4} -^{4} {C_{1}}^{303} C_{4} +^{4} {C_{2}}^{202} C_{4} -^{4} {C_{3}}^{101} C_{4}$ is equal to

Moderate

A

(401)⁴
B

(101)⁴
C

0
D

(201)⁴

Solution

The given expression is the coefficient of x⁴ in
$^{4} C_{0} (1 + x)^{404} -^{4} C_{1} (1 + x)^{303} +^{4} C_{2} (1 + x)^{202} -^{4} C_{3} (1 + x)^{101} +^{4} C_{4}$
$\begin{matrix} = Coefficient of x^{4} in {[(1 + x)^{101} - 1]}^{4} \\ = Coefficient of x^{4} in {(^{101} C_{1} x +^{101} C_{2} x^{2} + \dots)}^{4} \\ = (101)^{4} \end{matrix}$

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