(A) The general term in the expansion of $(1 + x)^n$ is given by $T_{r+1} = ^nC_r x^r$.For the expansion of $(1 + x)^{p+q}$,the coefficient of $x^p$ i

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(A) The general term in the expansion of $(1 + x)^n$ is given by $T_{r+1} = ^nC_r x^r$.For the expansion of $(1 + x)^{p+q}$,the coefficient of $x^p$ is $^{p+q}C_p$.Similarly,the coefficient of $x^q$ is $^{p+q}C_q$.Using the property of binomial coefficients,$^nC_r = ^nC_{n-r}$,we have:$^{p+q}C_p = ^{p+q}C_{(p+q)-p} = ^{p+q}C_q$.Therefore,the coefficients of $x^p$ and $x^q$ are equal.

Class 11 Mathematics Binomial Theorem Properties of binomial coefficients

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If $p$ and $q$ are positive integers,then the coefficients of $x^p$ and $x^q$ in the expansion of $(1 + x)^{p + q}$ are

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Equal
B
Equal in magnitude but opposite in sign
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Reciprocal to each other
D
None of these

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The ratio of the coefficients of the terms $x^{n-r}a^r$ and $x^ra^{n-r}$ in the binomial expansion of $(x+a)^n$ is:

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If $(1+x)^n = p_0 + p_1 x + p_2 x^2 + \ldots + p_n x^n$,then the value of $p_0 + p_3 + p_6 + \ldots$ is equal to:

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If $(1 + x)^n = C_0 + C_1x + C_2x^2 + .......... + C_nx^n$,then $\frac{C_1}{C_0} + \frac{2C_2}{C_1} + \frac{3C_3}{C_2} + .... + \frac{nC_n}{C_{n - 1}} = $

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If $1^2 \cdot \binom{15}{1} + 2^2 \cdot \binom{15}{2} + 3^2 \cdot \binom{15}{3} + \ldots + 15^2 \cdot \binom{15}{15} = 2^m \cdot 3^n \cdot 5^k$, where $m, n, k \in N$, then $m + n + k$ is equal to:

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Let $S = \frac{1}{25!} + \frac{1}{3!23!} + \frac{1}{5!21!} + \dots$ up to $13$ terms. If $13S = \frac{2^{k}}{n!}$ where $k \in N$, then $n + k$ is equal to

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If $p$ and $q$ are positive integers,then the coefficients of $x^p$ and $x^q$ in the expansion of $(1 + x)^{p + q}$ are

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The ratio of the coefficients of the terms $x^{n-r}a^r$ and $x^ra^{n-r}$ in the binomial expansion of $(x+a)^n$ is:

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