Asif, Sean, Bill, Francis, and Andrew are running against each other in a 100 m race. They all have an equal chance of winning.
Show the possible orders for the first three runners crossing the finish line.
What is the probability of Andrew being one of the first three runners to cross the finish line? Assume that there will not be a tie.

You are asked to list the possible permutations of 5 runners, taken 3 at a time. Then, you are asked for the fraction of those that have Andrew among them.

Permutations

The formula for the number of permutations of n objects taken k at a time is ...

P(n, k) = n!/(n -k)!

Then the number of ways three can place from five contestants is ...

P(5, 3) = 5!/(5 -3)! = 5·4·3 = 60

Listing

For listing purposes, we can represent the runners with the letters ...

A: Andrew
B: Bill
C: Francis
D: Asif
E: Sean

The order we choose for listing the permutations is to list a combination of three finishers, followed by the other 5 orders in which they can finish. In order of first, second, third, we can have ...

ABC ACB BAC BCA CAB CBA ABD ADB BAD BDA DAB DBA

ABE AEB BAE BEA EAB EBA ACD ADC CAD CDA DAC DCA

ACE AEC CAE CEA EAC ECA ADE AED DAE DEA EAD EDA

BCD BDC CBD CDB DBC DCB BCE BEC CBE CEB EBC ECB

BDE BED DBE DEB EBD EDB CDE CED DCE DEC ECD EDC

P(Andrew shows)

In the above list, the first three of the five lines all have Andrew as one of those who "show" in the race.

P(Andrew shows) = 3/5 = 0.6