Question

Compute the number of possible raceout comes given the following scenarios:

a) How many different ways can a first, second and third place winner occur if there are five runners competing and no ties are allowed?

b) Now consider the general scenario that there are n runners in total and m distinct slots for which they are competing with no ties. How many different race outcomes are possible, in terms of n and m?

c) Now consider that the m slots for which they are competing are not distinct (i.e., there is an m-way tie for first-place and no other awards). How many different race outcomes are possible, in terms of n and m?

Answer 1

(a)

Number of ways of first, second and third winner for 5 runners is equivalent to Permutation of 3 items from 5 items as follows:

Answer is:

60

(b)

(a)

Number of ways of m distinct slots for n runners is equivalent to Permutation of m items from n items as follows:

Answer is:

(c)

Number of ways of m non-distinct slots for n runners is equivalent to Combination of m items from n items as follows:

So,

Answer is: