I mixed up all my keys!

In the worst case scenario:

In order to match your first key to the correct padlock, you need at most 100 tries. In order to match your second key to the correct padlock, you need at most 99 tries. Therefore, we need to calculate the value of 100 + 99 + 98 + … + 1 + 0, which is 5050.

Let

• $N$ be the number of keys and padlocks.
• $T$ be the number, at most, you need to try

$N=1$

You don't need to try, the key must match the padlock,

$T=0$

$N=2$

You pick up a key and try a padlocks fail. The key must match the other padlock and it remains 1 key and 1 padlock, i.e. N=1

$\therefore T=1 +0$

By induction, for any given $N > 0$ , we can prove $T = \sum\limits_{i=0}^{N-1} i$

For $N = 101$

$T = \sum\limits_{i=0}^{100} i = 5050$