Question

For $2\le r\le n$, ${}^{n}C_r+2{}^{n}C_{r+1}+{}^{n}C_{r+2}=$

• A. ${}^{n+1}C_{r-1}$
• B. $2{}^{n+1}C_{r+1}$
• C. $2{}^{n+2}C_r$
• D. ${}^{n+2}C_{r+2}$

Answer 1

The correct option is D

${}^{n+2}C_{r+2}$

Finding the value:

Given, $r\&n$ are positive integer, $2\le r\le n$

We know that,

${}^{\mathbf{n}}\mathit{C}_{\mathbf{r}}\mathbf{+}{}^{\mathbf{n}}\mathit{C}_{\mathbf{r}\mathbf{+}\mathbf{1}}\mathbf{=}{}^{\mathbf{n}\mathbf{+}\mathbf{1}}\mathit{C}_{\mathbf{r}\mathbf{+}\mathbf{1}}$

Using the same for given equation,

$\begin{array}{rcl}& & {}^{n}C_r+2{}^{n}C_{r+1}+{}^{n}C_{r+2}\\ & =& {}^{n}C_r+{}^{n}C_{r+1}+{}^{n}C_{r+1}+{}^{n}C_{r+2}\\ & =& {}^{n+1}C_{r+1}+{}^{n+1}C_{r+2}\\ & =& {}^{n+2}C_{r+2}\end{array}$

Hence the correct option is $\mathbf{\left(}\mathit{D}\mathbf{\right)}$