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Composite Function

If p and q ar...

Question

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

A

0

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B

p,

when

p > q

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C

q,

when

p < q

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D

| p - q |

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Solution

The correct option is A $0$
Given : $(1 + x)^{p + q}$
General term is
$T_{r + 1} =^{p + q} C_{r} (1)^{p + q - r} x^{r} \Rightarrow T_{r + 1} =^{p + q} C_{r} x^{r}$
Coefficient of $x^{p} =^{p + q} C_{p}$
Coefficient of $x^{q} =^{p + q} C_{q}$
So, $^{p + q} C_{p} =^{p + q} C_{q}$
Hence, the absolute difference between the coefficients is $0.$

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Composite Function

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