A) \[\left( \begin{matrix} n+1 \\ r-1 \\ \end{matrix} \right)\]
B) \[2\,\left( \begin{matrix} n+1 \\ r+1 \\ \end{matrix} \right)\]
C) \[2\,\left( \begin{matrix} n+2 \\ r \\ \end{matrix} \right)\]
D) \[\left( \begin{matrix} n+2 \\ r \\ \end{matrix} \right)\]
Correct Answer: D
Solution :
Expression \[={{\,}^{n}}{{C}_{r}}+2\,.{{\,}^{n}}{{C}_{r-1}}{{+}^{n}}{{C}_{r-2}}\] \[={{(}^{n}}{{C}_{r}}+{{\,}^{n}}{{C}_{r-1}})+{{(}^{n}}{{C}_{r-1}}+{{\,}^{n}}{{C}_{r-2}})\] \[={{\,}^{n+1}}{{C}_{r}}+{{\,}^{n+1}}{{C}_{r-1}}={{\,}^{n+2}}{{C}_{r}}\].
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