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

Question

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 (A) n(n+1) (B) (n(n-1)/2) (C) (n(n+1)/2) (D) None of these

Tardigrade Admin · Accepted Answer

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) + \dots + 1 = \frac{n ( n + 1 )}{2} =^{n + 1} C_{2}

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

$n (n + 1)$

$\frac{n ( n - 1 )}{2}$

$\frac{n ( n + 1 )}{2}$

None of these

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
$= n + (n - 1) + \dots + 1 = \frac{n ( n + 1 )}{2} =^{n + 1} C_{2}$

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

n(n+1)

2n(n−1)​

2n(n+1)​