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.)

Short answer

The required number of ways are\(75\).

Step-by-step solution

Definition of Sum rule, permutation and combination

The sum rule in probability is also known as the addition law of probability. It can be explained as a rule stating that an event's probability to occur among two events will be equal to the sum of probability of each event which will occur minus the probability that both events will occur. The generalized sum rule is formulated as\({\rm{P}}({\rm{A}} \cup {\rm{B}}) = {\rm{P}}({\rm{A}}) + {\rm{P}}({\rm{B}}) - {\rm{P}}({\rm{A}} \cap {\rm{B}})\).

Definition permutation (order is important):\(P(n,r) = \frac{{n!}}{{(n - r)!}}\)

Definition combination (order is not important):\(C(n,r) = \left( {\begin{array}{*{20}{l}}n\\r\end{array}} \right) = \frac{{n!}}{{r!(n - r)!}}\)

with \(n! = n \cdot (n - 1) \cdot \ldots \cdot 2 \cdot 1\)

Use permutation and division rule

For there to be no ties the number of permutations is\({\rm{P}}(4,4) = 4! = 24\).

For two horses tie, by the product rule, there are\({6^*}6 = 36\)possibilities for this case.

For two groups of two horses to tie there are\(C(4,2) = 6\)ways to choose the two winning horses The other two horses tie for second place.

For three horses to 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

For all four horses to tie there is only one combination for this.

By the sum rule, the total is \(24 + 36 + 6 + 8 + 1 = 75\). Therefore, the required number of ways are 75.