The coefficient of x^5 in the expansion of (1 | vedclass

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(C) The given expression is a geometric series: $S = (1 + x)^{21} + (1 + x)^{22} + \dots + (1 + x)^{30}$.Here,the first term $a = (1 + x)^{21}$,the co

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$^{31}C_6 - ^{21}C_6$

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(C) The given expression is a geometric series: $S = (1 + x)^{21} + (1 + x)^{22} + \dots + (1 + x)^{30}$.Here,the first term $a = (1 + x)^{21}$,the common ratio $r = (1 + x)$,and the number of terms $n = 10$.Using the sum formula $S = a \frac{r^n - 1}{r - 1}$,we get:$S = (1 + x)^{21} \left[ \frac{(1 + x)^{10} - 1}{(1 + x) - 1} \right] = \frac{1}{x} [(1 + x)^{31} - (1 + x)^{21}]$.To find the coefficient of $x^5$ in $S$,we need the coefficient of $x^5$ in $\frac{1}{x} [(1 + x)^{31} - (1 + x)^{21}]$.This is equivalent to finding the coefficient of $x^6$ in $[(1 + x)^{31} - (1 + x)^{21}]$.The coefficient of $x^6$ in $(1 + x)^{31}$ is $^{31}C_6$ and in $(1 + x)^{21}$ is $^{21}C_6$.Thus,the required coefficient is $^{31}C_6 - ^{21}C_6$.

The coefficient of $x^5$ in the expansion of $(1 + x)^{21} + (1 + x)^{22} + \dots + (1 + x)^{30}$ is

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