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Question

In the expansion of $$ (1+x)^n $$ the coefficients of $$p^{th}$$ and $$(p+1)^{th}$$ terms are respectively $$p$$ and $$q$$ then $$ p+q = $$


A
n
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B
n+1
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C
n+2
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D
n+3
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Solution

The correct option is B $$n+1$$
Binomial Coefficient of $$p^{th}$$ term $$\:^nC_{p-1}=p$$
Binomial Coefficient of $$(p+1)^{th}$$ term $$\:^nC_{p}=q$$
Therefore, with respect to the above question.
$$\dfrac{\:^nC_{p-1}}{\:^nC_{p}}=\dfrac{p}{q}$$
$$\dfrac{n!(n-p)!p!}{n!(n-p+1)!(p-1)!}=\dfrac{p}{q}$$
$$\dfrac{p}{n-p+1}=\dfrac{p}{q}$$
$$n-p+1=q$$
$$n=p+q-1$$
$$n+1=p+q$$
Hence answer is $$n+1$$

Mathematics

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