A) \[^{n+m+1}{{C}_{n+1}}\]
B) \[^{n+m+2}{{C}_{n}}\]
C) \[^{n+m+3}{{C}_{n-1}}\]
D) none of these
Correct Answer: A
Solution :
\[\sum\limits_{r=0}^{m}{^{n+r}{{C}_{n}}}=\sum\limits_{r=0}^{m}{^{n+r}{{C}_{r}}}\] \[{{=}^{n}}{{C}_{0}}{{+}^{n+1}}{{C}_{1}}{{+}^{n+2}}{{C}_{2}}+.....{{+}^{n+m}}{{C}_{m}}\] \[{{=}^{n+2}}{{C}_{2}}{{+}^{n+3}}{{C}_{3}}+...{{+}^{n+m}}{{C}_{m}}\] \[{{=}^{n+m}}{{C}_{m-1}}{{+}^{n+m}}{{C}_{m}}\] \[{{=}^{n+m+1}}{{C}_{m}}{{=}^{n+m+1}}{{C}_{n+1}}\]
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