Question:
If ${ }^{n} C_{r-1}={ }^{n} C_{3 r}$, find $r$.
Solution:
Given: ${ }^{n} \mathrm{C}_{r-1}={ }^{n} \mathrm{C}_{3 r}$
To find: $r=?$
We know that:
${ }^{n} C_{r}={ }^{n} C_{n-r}$
$\Rightarrow{ }^{n} C_{r-1}={ }^{n} C_{n-(r-1)}$
$\Rightarrow{ }^{n} C_{r-1}={ }^{n} C_{n-r+1}$
$\Rightarrow{ }^{n} C_{n-r+1}={ }^{n} C_{3 r}$
$\Rightarrow n-r+1=3 r$
$\Rightarrow 4 r=n+1$
⇒ $r=\frac{n+1}{4}$
Ans: $r=\frac{n+1}{4}$
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