Question:
If nC4 = nC6, find 12Cn.
Solution:
We have,
${ }^{n} C_{4}={ }^{n} C_{6}$
$\Rightarrow n=6+4=10 \quad\left[\because{ }^{n} C_{x}={ }^{n} C_{y} \Rightarrow x=y\right.$ or, $\left.n=x+y\right]$
Now, ${ }^{12} C_{10}={ }^{12} C_{2} \quad\left[\because{ }^{n} C_{r}={ }^{n} C_{n-r}\right]$
$\Rightarrow^{12} C_{10}={ }^{12} C_{2}=\frac{12}{2} \times \frac{11}{1} \times{ }^{10} C_{0} \quad\left[\because{ }^{n} C_{r}=\frac{n}{r}{ }^{n-1} C_{r-1}\right]$
$\Rightarrow{ }^{12} C_{10}=66 \quad\left[\because{ }^{n} C_{0}=1\right]$