The coefficient of $$x53$$ in ∑$$m=0100$$ $$100Cm(x$$−$$5)100$$−$$m(4)m$$ is

Binomial theorem for positive integral Index

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Question

The coefficient of $x^{53}$ in $\sum_{m = 0}^{100}^{100} C_{m} (x - 5)^{100 - m} (4)^{m}$ is

Moderate

Solution

We have
$\sum_{m = 0}^{100}^{100} C_{m} (x - 5)^{100 - m} (4)^{m} = (x - 5 + 4)^{100} = (x - 1)^{100}$
$∴$ Coefficient of $x^{53}$ is $^{100} C_{53} (- 1)^{47} = -^{100} C_{53}$

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