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.)

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Annie Levasseur · Accepted Answer

1. No ties:  The number of permutations is P(4,4)=4≠24  2. Two horses tie:  There are C(4,2)=6 ways to choose the two horses that tie. There are P(3,3)=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⋅6=36 possibilities for this case  3. Two groups of two horses tie  There are C(4,2)=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(4,3)=4 ways to choose the two horses that tie. There are P(2,2)=2 ways for the groups to finish. By the product rule, there are 4⋅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 · Accepted Answer

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karton · Accepted Answer

Result can be like this: A, B, C, D are horses and 1-4 are the final positions. 1 2 3 4 - 4! = 24 ways 1 2 3 3 - 4!/2! = 12 ways 1 2 2 3 - 4!/2! = 12 ways 1 2 2 2 - 4!/3! = 4 ways 1 1 2 3 - 12 ways 1 1 2 2 - 4!/2!2! = 6 ways 1 1 1 2 - 4 ways 1 1 1 1 - 1 way So, total = 75 ways

How many ways are there for a horse race with

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