Question

${}^m{C}_{r+1}+{\sum _{k=m}^n}^k{C}_r$ is equal to

• A. ${}^n{C}_{r+1}$
• B. ${}^{n+1}{C}_{r+1}$
• C. ${}^n{C}_r$
• D. None of these

Answer 1

The correct option is B

${}^{n+1}{C}_{r+1}$

Explanation for the correct option:

Sum of binomial coefficcient:

Given - to find the value of ${}^m{C}_{r+1}+{\sum _{k=m}^n}^k{C}_r$

$$\begin{gathered} {}^m{C}_{r+1}+{\sum _{k=m}^n}^k{C}_r \\ {=}^m{C}_{r+1}+\left[{}^m{C}_r{+}^{m+1}{C}_r{+}^{m+2}{C}_r+...{+}^n{C}_r\right] \\ =\left({}^m{C}_{r+1}{+}^m{C}_r\right){+}^{m+1}{C}_r{+}^{m+2}{C}_r+...{+}^n{C}_r \\ {=}^{m+1}{C}_{r+1}{+}^{m+1}{C}_r{+}^{m+2}{C}_r+...{+}^n{C}_r \\ {=}^{m+2}{C}_{r+1}{+}^{m+2}{C}_r+...{+}^n{C}_r \\ {=}^n{C}_{r+1}{+}^n{C}_r \\ {=}^{n+1}{C}_{r+1} \end{gathered}$$ $\left[{\because }^m{C}_r{+}^m{C}_{r+1}{=}^{m+1}{C}_{r+1}\right]$

Hence, the correct option is (B)