 Linda Seales

2022-01-07

How many ways are there for a horse race with four horses to finish if ties are possible? [Note: Any number of the four horses may tie.) Annie Levasseur

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1. No ties:
The number of permutations is $P\left(4,4\right)=4\ne 24$
2. Two horses tie:
There are $C\left(4,2\right)=6$ ways to choose the two horses that tie. There are $P\left(3,3\right)=6$ ways for the groups to finish A group is either a single horse or the two tying horses. By the product rule, there are $6\cdot 6=36$ possibilities for this case
3. Two groups of two horses tie
There are $C\left(4,2\right)=6$ ways to choose the two winning horses. The other two horses tie for second place.
4.Three horses tie with each other:
There are $C\left(4,3\right)=4$ ways to choose the two horses that tie. There are $P\left(2,2\right)=2$ ways for the groups to finish. By the product rule, there are $4\cdot 2=8$ possibilities for this case.
5. All four horses tie:
There is only one combination for this. By the sum rule, the total is $24+36+6+8+1=75$. ol3i4c5s4hr

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