Question

If ${}^{n}C_{12}={}^{n}C_6$, then ${}^{n}C_2$ is equal to

• A. $172$
• B. $153$
• C. $306$
• D. $2556$

Answer 1

The correct option is B

$153$

Explanation for the correct option

Given that ${}^{n}C_{12}={}^{n}C_6$

By the properties of ${}^n{C}_r$ we know that

If ${}^n{C}_x={}^n{C}_y$, then $x=y$ or $x+y=n$

$6\ne 12$$\Rightarrow$$6+12=n$

$\Rightarrow$$n=18$

$\therefore$${}^{n}C_2={}^{18}C_2$

$=\frac{18!}{16!2!}$ $\left[\because {}^{n}C_r=\frac{n!}{r!\left(n-r\right)!}\right]$

$=\frac{18\times 17}{2}$

$\Rightarrow$${}^{n}C_2=153$

Thus the value of ${}^{n}C_2$ is $153$.

Hence option(B) i.e. $153$ is the correct answer.