The coefficient of x^{53} in the expansion su | vedclass

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Question

(C) The given expression is of the form $\sum_{m=0}^{n} {^{n}C_m a^{n-m} b^m}$,which is the binomial expansion of $(a+b)^n$.Here,$a = (x-3)$,$b = 2$,a

Vedclass · Accepted Answer

$-^{100}C_{53}$

Vedclass · Answer

(C) The given expression is of the form $\sum_{m=0}^{n} {^{n}C_m a^{n-m} b^m}$,which is the binomial expansion of $(a+b)^n$.Here,$a = (x-3)$,$b = 2$,and $n = 100$.Therefore,the expression simplifies to $((x - 3) + 2)^{100} = (x - 1)^{100}$.We can rewrite this as $(-(1 - x))^{100} = (1 - x)^{100}$.The general term in the expansion of $(1 - x)^{100}$ is $T_{r+1} = {^{100}C_r (1)^{100-r} (-x)^r} = {^{100}C_r (-1)^r x^r}$.To find the coefficient of $x^{53}$,we set $r = 53$.The term is ${^{100}C_{53} (-1)^{53} x^{53}} = -{^{100}C_{53}} x^{53}$.Thus,the coefficient of $x^{53}$ is $-{^{100}C_{53}}$.

The coefficient of $x^{53}$ in the expansion $\sum_{m = 0}^{100} {^{100}C_m (x - 3)^{100 - m} \cdot 2^m}$ is

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