Question

If ⁿC₄ = ⁿC₆, find ¹²C_n.

Answer 1

We have,
${}^n{C}_4={}^n{C}_6$
$\Rightarrow n=6+4=10$ [∵ ${}^n{C}_x={}^n{C}_y\Rightarrow x=y$ or, $n=x+y$]

Now, ${}^{12}{C}_{10}={}^{12}{C}_2$ [∵ ${}^n{C}_r={}^n{C}_{n-r}$]
${\Rightarrow }^{12}{C}_{10}{=}^{12}{C}_2=\frac{12}{2}\times \frac{11}{1}{\times }^{10}{C}_0$ [∵ ${}^n{C}_r=\frac{n}{r}{}^{n-1}{C}_{r-1}$]
$\Rightarrow {}^{12}{C}_{10}=66$ [∵ ${}^n{C}_0=1$]