Math CentralQuandaries & Queries


Question from Heidi, a parent:

how many different combinations of 4 letters can be made from a set of 8 letters?

Hi Heidi,

Suppose for example the 8 letters are the first 8 letters of the alphabet, a to h. In mathematics the word combination indicates that order is unimportant, thus abcd and bdca are the same combination.

First let's look at making a "word" of 4 letters from the letters a to h where you can only use a letter once. To write down such a word you start with the first letter which can be any of the 8 letters available. Once you have chosen the first letter you have 7 choices for the second letter and hence 8 × 7 = 56 ways to construct a 2 letter word. Again, once you have the first two letters chosen you can choose the third letter in 6 ways giving 8 × 7 × 6 possible 3 letter words and similarly 8 × 7 × 6 × 5 = 1680 possible 4 letter words.

If order is not important then this list is too long since it contains the words abcd, dbca, bcad and so on. In fact, using the same argument as above, there are 4 × 3 × 2 × 1 = 24 possible words from the letters a, b, c and d. The same is true for any four distinct letters. Any choice of four distinct letters gives rise to 24 words. Thus the list of words is 24 times too large. Hence the number of combinations of 4 letters from 8 letters is 1680/24 = 70.

I hope this helps,

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