Math CentralQuandaries & Queries


Question from Annie, a student:

I want to know how many combinations are there for the first 3 horses in a race of 20 horses. For example, the horses can come in 1, 5, 7 or 7, 5, 1 which are totally different combos. I've only found formulas where the 1, 2, 3 and 3, 2, 1 are counted as the same combo.
In horse racing every combo is different.
Thank you.

Hi Annie,

You know how to calculate the number of ways of choosing 3 horses from 20 horses. Suppose this number is C. As you said this doesn't distinguish the order of the finish, just the number of ways to choose the first three horses. Once you have the first three horses, how many ways can the be arranged as first, second and third? For example suppose that the horses who finish in the first three are 3 , 2 and 7.

If 7 is first then they could finish 7, 2, 3 or 7, 3 ,2.
If 3 is first then they could finish 3, 2, 7 or 3, 7 ,2.
If 2 is first then they could finish 2, 3, 7 or 2, 7, 3.

Thus each of the C ways of choosing the first three finishers can be arranged into 6 different finishing orders for these 3 horses. Hence the number you want is 6 × C.

I hope this helps,

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