Question

In how many ways can a five horse race end, allowing for the possibility that two horses tie?

Answer 1

Answer: 360 ways.

Given: n =5 horses.

A. No ties:

The number of of ways race can end if there are no ties =P(5, 5) =5! =120

[or 1 winning horse out of 5 can be selected in C(5, 1) =5 ways and the remaining 4 horses can be arranged in P(4,4) =4! =24 ways. Thus, number of ways the race can end with no ties =5*24 =120]

B. 2 horses tie:

2 horses out of 5 horses can be selected in C(5, 2) ways. Now, there are (5) - (2 tied horses) + (1 entity of tied horses) =5-2+1 =4 sets/groups of horses which can be arranged in P(4, 4) =4! ways.

So, the number of ways the race can end with 2 horses tie =C(5, 2)*P(4, 4) =10*24 =240

Therefore, the total number of ways a five horse race can end allowing for 2 horses tie =120 + 240 =360 ways.