Question:
If ${ }^{20} C_{r+1}={ }^{20} C_{r-10}$ then find the value of ${ }^{10} C_{r}$.
Solution:
Given: ${ }^{20} \mathrm{C}_{r+1}={ }^{20} \mathrm{Cr}-10$ Need to find: Value of ${ }^{10} \mathrm{Cr}$ We know, one of the
property of combination is: If ${ }^{n} C_{r}={ }^{n} C_{t}$, then,
(i) $r=t$ OR
(ii) $r+t=n$ We can't apply the property
(i) here. So we are going to use property
(ii) ${ }^{20} \mathrm{Cr}+1={ }^{20} \mathrm{Cr}-10 \mathrm{By}$ the property
(ii), $\Rightarrow r+1+r-10=20 \Rightarrow 2 r=29 \Rightarrow r=14.5$. We need to find out the value of ${ }^{10} \mathrm{Cr}$. But here $r$ can't be a rational number. Therefore the value of ${ }^{10} C_{r}$ can't be find out.