Combinations and horse races

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Question from Ian:

How do I calculate the number of combinations for:
Horse races,1 horse in each race.I want all combinations covered.
e.g.5 races, 1 horse in each race. R 1&2, R 1&3,R 1&4, R 2,3,4,5, R 1,2,3,4,5 etc etc.I have worked it out manually to 27 combinations but if I want to cover more than 5 it will be a big excercise to work out the cost manually.
Many Thanks

Ian,

I am not completely sure I understand the question, but maybe this will help. I think you are looking for the number of subsets of races that each contain at least two races. I'll just call these collections of races instead of subsets.

Since each race is either in, or not in, a particular collection of races, there are 2⁵ = 32 possibilities. This includes the collection that contains no races. In general if you have n races, there are 2ⁿ possibilities.

Among the 32, there is one collection that contains no races and there are 5 that each contain a single race. This means that there are 32-6 = 26 collections of two or more races. When there are n races, there are 2ⁿ - 1 - n such
collections.

Victoria

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