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