Daily Question 114

Daily Question 114

$\textbf{Q114.}$ The value of $\displaystyle\sum_{k=0}^{7}\left[\dfrac{\displaystyle{7 \choose k}}{\displaystyle{14 \choose k}}\sum_{r=k}^{14}{r \choose k}{14 \choose r}\right]$ where $\displaystyle{n \choose x}$ denotes ${^{n}}C_{r}$, is

(A) $6^7$

(B) $8^7$

(C) greater than $7^6$

(D) greater than $7^{8}$

$\textbf{Ans.} (A), (C)$

$\textbf{Sol.}$ Given $\displaystyle\sum_{k=0}^{7}\left[\dfrac{\displaystyle{7 \choose k}}{\displaystyle{14 \choose k}}\sum_{r=k}^{14}{r \choose k}{14 \choose r}\right]$

$=\displaystyle\sum_{k=0}^{7}\left[\dfrac{{^{7}}C_k}{{^{14}C_k}}\sum_{r=k}^{14}{^{r}}C_k\cdot{^{14}}C_{r}\right]$

$=\displaystyle\sum_{k=0}^{7}\left[{^{7}}C_k\dfrac{k!(14-k)!}{14!}\sum_{r=k}^{14}\dfrac{r!}{k!(r-k!)}\cdot\dfrac{14!}{r!(14-r)!}\right]$

$=\displaystyle\sum_{k=0}^{7}\left[{^{7}}C_k\sum_{r=k}^{14}{^{14-k}}C_{r-k}\right]$

$=\displaystyle\sum_{k=0}^{7}{^{7}}C_k\cdot 2^{14-k}$

$=2^{14}\displaystyle\sum_{k=0}^{7}{^{7}}C_k\left(\dfrac{1}{2}\right)^{k}$

$=2^{14}\cdot\left(1+\dfrac{1}{2}\right)^{7}$

$=6^{7} > 7^6$

Daily Question 113

Compiled Daily Questions 1-100