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Question:
Given the integers $r>1, n>2$, and coefficients of $(3 r)^{\text {th }}$ and $(r+2)^{\text {nd }}$ terms in the binomial expansion of $(1+x)^{2 n}$ are equal, then
(A) $n=2 r$
(B) $n=3 r$
(C) $n=2 r+1$
(D) none of these
Solution:
(A) $n=2 r$
Explanation:
Given $(1+x)^{2 n}$
$T_{3 r}=T_{(3 r-1)+1}={ }^{2 n} \mathrm{C}_{3 r-1} x^{3 r-1}$
$T_{r+2}=T_{(r+1)+1}={ }^{2 n} C_{r+1} x^{r+1}$
${ }^{2 n} C_{3 r-1}={ }^{2 n} C_{r+1}$
$3 r-1+r+1=2 n$
$n=2 r$
Hence option A is the correct answer.
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