Question

For fundraising, the local elementary school puts on a jog-a-thon where students will raise money by jogging as many miles as possible and collecting donations from sponsors. At the end of the jog-a-thon, the top 4 joggers (the four that ran the most miles) will be given awards. On the day of the jog-a-thon, 60 kids show up to participate.

a) If all four winners were given a $35 gift card, how many unique group winners could be formed at the end of the race?

b) Suppose instead that each of the top four finishers were given different prizes: $50 for first place, $40 for second place. $30 for third place, $20 for fourth place. How many unique ways could the prizes be assigned in this scenario?

c) Suppose instead all participants put their name in a bowl, four are randomly chosen to receive a gift card, and students can win more than one of the four prizes. How many unique ways can the prizes be given out in this scenario?

Answer 1

a)

Since all four winners get the same prize so order of winners is not important.

That is number of ways of selecting 4 joggers out of 60 is

That is number of unique group winners could be formed at the end of the race is 487635.

b)

Now we have different prizes for different positions.

The number of ways of selecting winner for first prize is 60, number of ways of selecting winner for second prize is 59, number of ways of selecting winner for third prize is 58 and number of ways of selecting winner for fourth prize is 57. So possible number of unique groups is

60*59*58*57 = 11703240

The number of unique group winners could the prizes be assigned in this scenario is 11703240.

c)

Now since students can win more than one of the four prizes so draws are with replacement. So number of ways of distributing prizes is

60*60*60*60 = 12960000