Question:
If nCr + nCr + 1 = n + 1Cx , then x =
(a) r
(b) r − 1
(c) n
(d) r + 1
Solution:
(d) r + 1
$n_{C_{r}}+n_{C_{r+1}}=n+1_{C_{x}} \quad$ [Given]
We have:
${ }^{n} C_{r}+{ }^{n} C_{r+1}={ }^{n+1} C_{x} \quad\left[\because{ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}\right]$
$\Rightarrow{ }^{n+1} C_{r+1}={ }^{n+1} C_{x}$$\Rightarrow r+1=x \quad\left[\because{ }^{n} C_{x}={ }^{n} C_{y} \Rightarrow n=x+y\right.$ or $\left.x=y\right]$
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