Combinations and Probability (8 Problems) - Camp Cawapati
1. At Camp Cawapati, there is a prize ceremony at the end of each camp session. The campers compete at a variety of challenges throughout the 2-week session, in hopes of earning trophies. For each challenge a camper wins, the camper receives a small replica of the Sacred Cawapati Cup. (These replica trophies are all identical.)

(a) In the junior camp (ages 6 and 7), the 12 young campers compete in 12 different challenges, with a single winner identified in each challenge. The camp counselors arrange it so that no child wins more than one of these challenges, to ensure that the prizes will be spread around as much as possible among the campers.

(i)In how many different ways can the 12 challenge winners be determined?

(ii) In how many different ways could the 12 trophies be awarded?

(iii) What is the probability that Bobby, one of the Junior campers, will win at least one trophy?

(b) In the intermediate camp (ages 8-12), things work a bit differently. On arrival at the camp, each of the campers is assigned to a particular "herd" and the competition is between herds, rather than between individuals. (Trophies are distributed to all members of a winning herd.) There are 8 herds (with 8 campers in each herd) and 12 challenges. There is only one winning herd per challenge, and there is no restriction on how many challenges a herd may win.

(i) In how many different ways can the 12 challenge winners (herds) be determined?

(ii) In how many different ways could the 12 sets of trophies be awarded?

(iii) If each herd is equally likely to win each challenge, what is the probability that the Black Angus Bulls, one of the herds, will win exactly 3 of the challenges?

(c) In Senior camp ( ages 13 and 14 ), the competition gets stiffer. There are 14 Seniors who compete individually in the 6 challenges. Once again there is no restriction on how many challenges any individual camper can win, and every session there's someone who thinks he or she can "Punch the Pati" and win all 6. (They rarely do, but if successful, the camper's name is inscribed on the original "Sacred Cawapati Cup" .)

(i) In how many different ways can the challenge winners be determined?

(ii) If each Senior is equally likely to win each of the 6 challenges, what is the probability that each challenge will be won by a different camper?

(iii) If each Senior is equally likely to win each challenge, what is the Probability that Hank, one of the Seniors, will win at least one challenge?