EXAMPLE 15 Four-Letter Words

Suppose you have the following four Scrabble tiles: T, S, O, and P. How many four-letter sequences can you make using all four tiles? The only way to make a four-letter sequence is to use each letter exactly once. So, there are no repeats. This is a permutation by Rule B with and . To think through the problem (rather than simply plugging into the formula), proceed like this: Any of the four letters can be chosen first, any of the three that remain can be chosen second, and so on. The number of permutations, therefore, is

This example shows us that the permutation of all elements of a collection yields the product of the first integers. This expression of factors is special enough to have its own name: factorial (discussed in Chapter 2, and again in Chapter 11).

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Now, let’s get back to the Scrabble game. It turns out that only 6 of the 24 possible sequences of 4 letters are actually words in the English language (can you figure out all 6 words?), so the probability that a permutation (chosen at random from these 4 tiles) I will be an actual word is .