Question

There are $n$ locks and $n$ matching keys. If all the locks and keys are to be perfectly matched, find the maximum number of trials required to open a lock.

• A. None of the above
• B. $\frac{n(n+1)}{2}$
• C. $\frac{n(n+1)(n+2)}{6}$
• D. $\frac{n(n-1)}{2}$

Answer 1

The correct option is B $\frac{n(n+1)}{2}$

The maximum number of trials needed for the first key is $n$. For the second key, it will be $n-1$.

Now, for the $r$ th key, the maximum number of trials needed is $n-r+1$. Thus, the required answer is
$$n+(n-1)+\cdots +1=\frac{n(n+1)}{2}$$