Question

For $2 \le \, r \, \le \, n,\binom{n}{r}+2 \binom{n}{r-1}+\binom{n+2}{r}$ is equal to

• A. $\binom{n+1}{r-1}$
• B. $2 \binom{n+1}{r+1}$
• C. $2 \binom{n+1}$
• D. $2 \binom{n+1}{r}$

Answer 1

The Correct Option is D

$\binom{n}{r}+2 \binom{n}{r-1}+\binom{n}{r-2}=\bigg[\binom{n}{r}+\binom{n}{r-1}\bigg]$
+$\bigg[\binom{n}{n-r}+\binom{n}{n-r}\bigg]=\binom{n+1}{r} + \binom{n+1}{r-1}=\binom{n+1}{r}$
$[\because \, ^nC_r + \, ^nC_{r-1}= \, ^{n+1}C_r]$