If the coefficient of the middle term in the

expansion of $(1 + x)^{2 n + 2}$ is p and the coefficients of

middle terms in the expansion of $(1 + x)^{2 n + 1}$ are q and r, then

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Solution

The correct option is C

Since $(n + 2)^{t h}$ term is the middle term in the

expansion of $(1 + x)^{2 n + 2}$ , therefore p = $^{2 n + 2} C_{n + 1}$ .

Since $(n + 1)^{t h}$ and $(n + 2)^{t h}$ terms are middle

terms in the expansion of $(1 + x)^{2 n + 1}$ , therefore

q = $^{2 n + 1} C_{n}$ and r = $^{2 n + 1} C_{2 n + 1}$ But

$^{2 n + 1} C_{n}$ + $^{2 n + 1} C_{n + 1}$ = $^{2 n + 2} C_{n + 1}$

∴ q + r = p

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If the coefficient of the middle term in the expansion of (1+x)2n+2 is p and the coefficients of middle terms in the expansion of (1+x)2n+1 are q and r, then

The correct option is C Since (n+2)th term is the middle term in the expansion of (1+x)2n+2, therefore p = 2n+2Cn+1. Since (n+1)th and (n+2)th terms are middle terms in the expansion of (1+x)2n+1, therefore q = 2n+1Cn and r =2n+1C2n+1 But 2n+1Cn +2n+1Cn+1 = 2n+2Cn+1 ∴ q + r = p

If the coefficient of the middle term in the

expansion of $(1 + x)^{2 n + 2}$ is p and the coefficients of

middle terms in the expansion of $(1 + x)^{2 n + 1}$ are q and r, then