Combinations, on the other hand, are pretty easy going. Alice, Bob and Charlie is the same as Charlie, Bob and Alice. A joke: A “combination lock” should really be called a “permutation lock”. The order you put the numbers in matters. A true “combination lock” would accept both 10-17-23 and 23-17-10 as correct.
Permutations: The hairy details Let’s start with permutations, or all possible ways of doing something. We’re going to use permutations since the order we hand out these medals matters. Clever how I made the names match up with letters, eh? Let’s say A wins the Gold. Silver medal: 7 choices: B C D E F G H.
Let’s say B wins the silver. Bronze medal: 6 choices: C D E F G H. We picked certain people to win, but the details don’t matter: we had 8 choices at first, then 7, then 6. We had to order 3 people out of 8. And why did we use the number 5? Because it was left over after we picked 3 medals from 8.
Use the first 3 numbers of 8! You can mix it up and it looks the same. Let’s say I’m a cheapskate and can’t afford separate Gold, Silver and Bronze medals. In fact, I can only afford empty tin cans. How many ways can I give 3 tin cans to 8 people? Well, in this case, the order we pick people doesn’t matter. If I give a can to Alice, Bob and then Charlie, it’s the same as giving to Charlie, Alice and then Bob.