Solve this


If ${ }^{n} C_{r-1}={ }^{n} C_{3 r}$, find $r$.



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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