If coefficient of (2r+3)th and (r−1)th terms in the expansion of (1+x)1

Question

If coefficient of $(2 r + 3)^{t h}$ and $(r - 1)^{t h}$ terms in the expansion of $(1 + x)^{15}$ are equal, then value of r is

Options:

A . 5

B . 6

C . 4

D . 3

Answer: Option A
:
A

$^{15} C_{2 r + 2}$ = $^{15} C_{r - 2}$

But $^{15} C_{2 r + 2}$ = $^{15} C_{15 - (2 r + 2)}$ = $^{15} C_{13 - 2 r}$

⇒ $^{15} C_{13 - 2 r}$ = $^{15} C_{r - 2}$ ⇒ r = 5.

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