Question:
If ${ }^{n} C_{r+1}={ }^{n} C_{8}$ then find the value of ${ }^{22} C_{n}$.
Solution:
Given: ${ }^{n} C_{r+1}={ }^{n} C_{8}$ Need to find: Value of ${ }^{22} C_{n}$ We know, one of the property of
combination is: If ${ }^{n} C_{r}={ }^{n} C_{t}$, then, (i) $r=t \mathrm{OR}$ (ii) $r+t=n$ We are going to use property
(i) ${ }^{n} C_{r+1}={ }^{n} C_{8}$ By the property (i), $\Rightarrow r+1=8 \Rightarrow r=7$ Now we are going to use property
(ii) $\Rightarrow \mathrm{n}=8+7+1=16$ Therefore, ${ }^{22} \mathrm{C}_{\mathrm{n}}={ }^{22} \mathrm{C}_{16}=74613$.