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Question

Mathematics Question on Binomial theorem

For 2≤ r ≤ n,(nr)+2(nr−1)+(n+2r)2 \le \, r \, \le \, n,\binom{n}{r}+2 \binom{n}{r-1}+\binom{n+2}{r} is equal to

A

(n+1r−1)\binom{n+1}{r-1}

B

2(n+1r+1)2 \binom{n+1}{r+1}

C

2 \binom{n+1}

D

2(n+1r)2 \binom{n+1}{r}

Answer

2(n+1r)2 \binom{n+1}{r}

Explanation

Solution

(nr)+2(nr−1)+(nr−2)=[(nr)+(nr−1)]\binom{n}{r}+2 \binom{n}{r-1}+\binom{n}{r-2}=\bigg[\binom{n}{r}+\binom{n}{r-1}\bigg]
+[(nn−r)+(nn−r)]=(n+1r)+(n+1r−1)=(n+1r)\bigg[\binom{n}{n-r}+\binom{n}{n-r}\bigg]=\binom{n+1}{r} + \binom{n+1}{r-1}=\binom{n+1}{r}
[∵ nCr+ nCr−1= n+1Cr][\because \, ^nC_r + \, ^nC_{r-1}= \, ^{n+1}C_r]