Kenneth Rosen Edition 7 Exercise 6.3 Question 44 (Page No. 415)

Question

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.)

Answer 1

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