Question

The least value of natural number $n$ satisfying $C\left(n,5\right)+C\left(n,6\right)>C\left(n+1,5\right)$ is?

• A. $11$
• B. $10$
• C. $12$
• D. $13$

Answer 1

The correct option is A

$11$

Calculate the value of $n$ based on given information

The given inequality is,

$C\left(n,5\right)+C\left(n,6\right)>C\left(n+1,5\right)$

$\Rightarrow$ ${}^{n}C_5+{}^{n}C_6>{}^{n+1}C_5$

$\Rightarrow$ ${}^{n}C_6+{}^{n}C_5>{}^{n+1}C_5$

$\Rightarrow$ ${}^{n+1}C_6>{}^{n+1}C_5$ …$\left[\because {}^{n}C_r+{}^{n}C_{r-1}={}^{n+1}C_r\right]$

$\Rightarrow$ $\frac{\left(n+1\right)!}{6!\left(n+1-6\right)!}>\frac{\left(n+1\right)!}{5!\left(n+1-5\right)!}$ …$\left[\because {}^{n}C_r=\frac{n!}{r!\left(n-r!\right)}\right]$

$\Rightarrow$ $\frac{\left(n+1\right)!}{6\times 5!\times \left(n+1-6\right)!}>\frac{\left(n+1\right)!}{5!\left(n+1-5\right)\times \left(n+1-6\right)!}$

$\Rightarrow$ $\frac{1}{6}>\frac{1}{n-4}$

$\Rightarrow$ $n-4>6$

$\Rightarrow$ $n>6+4$

$\Rightarrow$ $n>10$

Thus, the least value of $n$ is $11$.

Hence, option A is the correct option.