Question:
If ${ }^{20} C_{r}={ }^{20} C_{r-10}$ then find the value of ${ }^{17} C_{r}$.
Solution:
Given: ${ }^{20} \mathrm{C}_{r}={ }^{20} \mathrm{C}_{r-10}$ Need to find: Value of ${ }^{17} \mathrm{C}_{r}$ 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 can't apply the property
(i) here. So we are going to use property (ii) ${ }^{20} C_{r}={ }^{20} C_{r-10}$ By the property
(ii), $\Rightarrow r+r-$ $10=20 \Rightarrow 2 r=30 \Rightarrow r=15$ Therefore, ${ }^{17} C_{15}=136$