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Let’s jump straight in. Suppose I have a collection of n different objects, where n is some unknown but fixed whole number. Suppose I want to arrange k of them in a row on my shelf, where k is some unknown but fixed whole number less than or equal to n. In how many ways can this be done?

This is a counting problem. You are being asked to count how many of something there are. The answer will be a formula involving the unknown quantities n and k. Counting is often thought of as being easy. While it is true that the things which must be counted are easy to describe (I have just done so in a couple of sentences), this does not mean that the problem of counting them is easy. So, in this post I continue my quest to explain some basic things about counting and how to do it.

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Now that you’ve decided on your outfit, you take a trip to the cinema with a friend. When you get there, you see that there are only two seats left and they are next to each other. In how many different ways can you and your friend occupy these two seats? Well, once seated, either you will be sitting on the left and your friend will be to the right of you or you are sitting on the right with your friend on your left. So there are two ways in total. This is very easy to see, but what if there were three of you and three seats left? In how many ways can you and your two friends occupy the three seats?

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