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Q.
If $\displaystyle\sum_{r=2}^n \frac{{ }^{n-2} C_{r-2}}{{ }^n C_r}=20$, then $n$ equals
Binomial Theorem
Solution:
$\displaystyle\sum_{ r =2}^{ n } \frac{{ }^{ n -2} C _{ r -2}}{\frac{ n ( n -1)}{ r ( r -1)} \cdot{ }^{ n -2} C _{ r -2}}=20$
$\displaystyle\sum_{ r =2}^{ n } \frac{ r ( r -1)}{ n ( n -1)}=20 $
$\frac{1}{ n ( n -1)} \displaystyle\sum_{ r =2}^{ n }\left( r ^2- r \right)=20 $
$\frac{1}{ n ( n -1)}\left(\frac{ n ( n +1)(2 n +1)}{6}-1-\left(\frac{ n ( n +1)}{2}-1\right)\right)=20 $
$\Rightarrow n =59 $