Question

$\sum _{r=0}^{m}n+r{c}_r$ is equal to

• (A) ^{n + m + 1}Cn + 1
• (B) ^{n + m + 2}Cn
• (C) ^{n + m + 3}Cn - 1
• (D) None of these

Answer 1

The Correct Option is A

Explanation:
$\begin{array}{rl}& \sum _{r=0}^{m}n+r_C\\ =& {}^{\pi }{C}_0{+}^{n+1}{C}_1{+}^{n+2}{C}_2{+}^{n+3}{C}_3+\dots \dots {+}^{n+m}{C}_m\\ =& {}^{n+3}{C}_2{+}^{n+3}{C}_3+\dots \dots {+}^{n+m}{C}_m\\ =& {}^{n+m}{C}_{m-1}{+}^{n+m}{C}_m\\ =& {}^{n+m+1}{C}_m{=}^{n+m+1}{C}_{n+1}\end{array}$