Question

The least positive integer $n$ for which ${}^{n-1}C_5+{}^{n-1}C_6<{}^{n}C_7$ is

• A. $14$
• B. $15$
• C. $16$
• D. $28$

Answer 1

The correct option is A

$14$

Solve the given expression

Given expression, ${}^{n-1}C_5+{}^{n-1}C_6<{}^{n}C_7$

Now to solve this we need to simplify this equation,

So, simplifying ${}^{n-7}C_5+{}^{n-1}C_6<{}^{n}C_7$, we get

$$\begin{gathered} {}^{n-1}C_5+{}^{n-1}C_7{<}^n{C}_7 \\ \Rightarrow \frac{\left(n-1\right)!}{5!\left(n-6\right)!}+\frac{\left(n-1\right)!}{6!\left(n-7\right)!}<\frac{n!}{7!\left(n-7\right)!} \\ \Rightarrow \frac{6\left(n-1\right)+\left(n-6\right)\left(n-1\right)!}{6!\left(n-6\right)\left(n-7\right)!}<\frac{n!}{7!\left(n-7\right)!} \\ \Rightarrow \frac{\left(n-1\right)!\left(6+n-6\right)}{6!\left(n-6\right)\left(n-7\right)!}<\frac{n!}{7!\left(n-7\right)!} \\ \Rightarrow \frac{1}{\left(n-6\right)}<\frac{1}{7} \\ \Rightarrow n-6<7 \\ \Rightarrow n>13 \end{gathered}$$

Therefore, the value least positive value of​ $n$ is $14$

Hence, option (A) is correct .