Keys and key hole puzzle odds.

Question

You have 13 keys... and 5 key holes....

5 keys work in the 5 key holes...

In random chance what are the odds of getting randomly all 5 keys in the right key holes on the first atempt.

Bonus. getting 4right and 1 wrong on on the first attempt. And getting the one wrong one correct on a second atempt

Are we to assume that exactly one key works in each of the five key holes? Or that each of five keys works in each of the five key holes? — Michael Joyce, Apr 21 '14 at 04:14

score 0 · Answer 1 · answered Apr 21 '14 at 08:49

Assuming that each of the 5 keys will unlock 1 specific lock, you can get the result in (at least) two different ways:

For each lock, choose a key at random: The chance that you choose the right key for the first lock is $\frac{1}{13}$. Choosing among the remaining keys, you will get the next key right with probability $\frac{1}{12}$, and so forth, giving you \begin{align*} \frac{1}{13}\cdot\frac{1}{12}\cdot\frac{1}{11}\cdot\frac{1}{10}\cdot\frac{1}{9}=\frac{1}{154440}. \end{align*}
Choosing 5 keys first, then place each in a lock: You can choose the 5 keys in $13\choose{5}$ ways. Afterwards, you can place each in a lock in $5!$ different ways. Only one of these are right, giving you a probability of \begin{align*} \frac{1}{{13\choose{5}} 5!}=\frac{1}{154440}. \end{align*}

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