If an the expansion of

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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