April 17th Problems

1. How many ways can two dice (six-sided) be rolled to get a 9?
2. How many ways can 4 dice by rolled to get a 6?
3. Andy, Brahbrah/Barbara, and Chuck each roll two dice (Monopoly maybe?). How many ways can Andy roll a 10, Brahbrah roll a 7, and Chuck a 5?
4. Suppose 13 men, 6 women, 2 boys, and 4 girls are the last survivors of the human race.
   1. How many ways can we film a 50s sitcom? That is, how many ways can they select 1 man, 1 woman, 1 boy, and 1 girl?
   2. How many ways can a man or a girl be selected? (This was the specified sacrifice (man or girl) for our new alien overlords.)
   3. How many ways can we select our new leader? (Select one human.)
5. Let \(A = \{a_1, \ldots, a_n\}\) and \(B = \{1, 2, 3\}\).
   1. Prove that there are \(3^n\) functions from \(A \to B\).
   2. How many of these functions are NOT onto?
   3. How many are onto?
6. Calculate \(3^{80} \mod {7}\).
7. Find the last 2 digits of \(7^{355}\).
8. How many cards of a single suit must exist in a set of \(n\) cards drawn from a standard deck of playing cards? (A deck of cards has 52 cards divided into 4 suits of 13 cards each. I'm looking for the number of guaranteed cards for any one suit without specifying which suit.)
9. How many ways can 10 adults and 5 children stand in a circle such that no two children are next to each other?