Rupert has three wooden letters: an 𝐴, a 𝐵, and a 𝐶. In how many different ways can he order all three letters?
To answer this question, we should consider how many choices Rupert has for each letter. These three spaces represent the three positions for the letter.
For the first letter, Rupert has all three letters available, so he has three choices. However, for the second letter, Rupert has already used one of the letters, so he only has two choices available. For the third letter, Rupert has already used two of the letters, so he only has one left. In fact, he doesn’t really have a choice at all. He has to use the last remaining letter.
Now what do we do with these numbers? As any of the options for the first letter can be combined with either of the options for the second and then the one option for the third, we multiply these numbers together to give the overall number of possibilities. Three multiplied by two multiplied by one is equal to six. So there are six different ways that Rupert can order all three letters.
In fact, as the number of letters in this question is quite small, we could also answer the question by listing out all of the possibilities. Rupert could start with an 𝐴 and then a 𝐵 and then a 𝐶. Or he could swap the 𝐵 and the 𝐶 around, giving 𝐴𝐶𝐵. These are the only two possibilities starting with the letter 𝐴. If Rupert starts with the letter 𝐵, he could have the word 𝐵𝐴𝐶, or he could swap the order of the 𝐴 and the 𝐶 around. These are the only two possibilities beginning with the letter 𝐵. Finally, we consider the possibilities beginning with the letter 𝐶. We could have 𝐶𝐴𝐵. Or once again, Rupert could swap the final two letters around to give the word 𝐶𝐵𝐴. And these are the only two possibilities beginning with the letter 𝐶.
So we’ve written out the full list of possibilities. And by counting them, we can see once again that the number of different ways in which Rupert can order all three letters is six. This method of listing out all of the possibilities worked in this case as we only had a small number of choices for each letter. If we had a larger number of choices, this wouldn’t really be realistic. So we’d have to remember the rule of multiplying the number of choices for each letter together to give the total number of possibilities.