Question:
If an the expansion of $(1+x)^{15}$, the coefficients of $(2 r+3)^{\text {th }}$ and $(r-1)^{\text {th }}$ terms are equal, then the value of $r$ is
(a) 5
(b) 6
(c) 4
(d) 3
Solution:
(a) 5
Coefficients of $(2 r+3)$ th and $(r-1)$ th terms in the given expansion are ${ }^{15} C_{2 r+2}$ and ${ }^{15} C_{r-2}$.
Thus, we have
${ }^{15} C_{2 r+2}={ }^{15} C_{r-2}$
$\Rightarrow 2 r+2=r-2 \quad$ or $2 r+2+r-2=15 \quad\left[\because\right.$ if ${ }^{n} C_{x}={ }^{n} C_{y} \Rightarrow x=y$ or $\left.x+y=n\right]$
$\Rightarrow r=-4 \quad$ or $r=5$
Neglecting the negative value, We have
$r=5$
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