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Q.
There are $n$ locks and $n$ matching keys. If all the locks and keys are to be perfectly matched, then maximum number of trails is equal to
Permutations and Combinations
Solution:
For the first key, maximum number of trials needed is $n$.
For second key it will be $(n-1)$.
In general, for $r$ th key, maximum number is needed trials is $(n-r+1)$.
Thus, required answers
$=n+(n-1)+\cdots+1=\frac{n(n+1)}{2}={ }^{n+1} C_{2}$