Solve the following

Question:

If nCr + nCr + 1 = n + 1Cx , then x =

(a) r

(b) − 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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