Problems

I have \(k\) different keys on my key ring. When I come home at night I try one key after another until I find the key that fits my front door. What is the probability that I find the correct key in exactly \(n\) attempts in each of the following three cases?

1. At each attempt, I choose a key that I have not tried before but otherwise each choice is equally likely.
2. At each attempt, I choose a key from all my keys and each of the \(k\) choices is equally likely.
3. At the first attempt, I choose from all my keys and each of the \(k\) choices is equally likely. Thereafter, I choose from the keys that I did not try the previous time but otherwise each choice is equally likely.