Solve the following

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, ${ }^{12} C_{10}={ }^{12} C_{2} \quad\left[\because{ }^{n} C_{r}={ }^{n} C_{n-r}\right]$

$\Rightarrow^{12} C_{10}={ }^{12} C_{2}=\frac{12}{2} \times \frac{11}{1} \times{ }^{10} C_{0} \quad\left[\because{ }^{n} C_{r}=\frac{n}{r}{ }^{n-1} C_{r-1}\right]$

$\Rightarrow{ }^{12} C_{10}=66 \quad\left[\because{ }^{n} C_{0}=1\right]$

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