## Combinations, Permutations and Probabilities

\[P(A)=\frac{Number \: of \: Possible \: Outcomes \: Including \: A)}{Total \: Number \: of Possible \: Outcome}\]

Suppose that a race has ten runners. Three fiends, Alf, Bob and Charles want to find the probability that they will take the finish first, second and third in some order.

The total number of ways they can take the first three places is

\[{}^{10}C_3\]

(choosing three from ten).The total number of possible ways we can order the finishing of the runners is

\[10!\]

.Hence

\[P(Alf, \: Bob \: and \: Charles \: Take \: First \: Three \: Places)=\frac{{}^{10}C_3}{10!}=\frac{(10!)/3!7!}{10!}=\frac{1}{3!7!}\]