Combinations Problems Problem 1: Sometimes we can use several counting techniques in the same problem, such as combinations and the addition principle. Say you are buying a sundae with one scoop of vanilla ice cream, and you have a choice of up to 3 toppings. In how many ways can you select your toppings? (Hint: Calculate the number of ways to choose no topping, C(3, 0), the number of ways to choose 1 topping, C(3, 1), 2 toppings, C(3, 2), and 3 toppings, C(3, 3). Then add together all the possibilities.)

Combinations Problems Problem 2: How many combinations are possible for 4 toppings? For 5 toppings? Based on your results, without calculating the answer what would you predict for the number of possible combinations with 6 toppings?

Combinations Problems Problem 3: You are about to take a 10-question true/false quiz. The teacher confides that exactly 3 of the questions are true. a.) In how many ways can you choose 3 questions to answer true? b.) In how many ways can you choose 7 questions to answer false? c.) In how many total possible ways can you answer the quiz, regardless of the teacher’s revelation about 3 questions being true?

Combinations Problems Problem 4: Imagine a state lottery has players choose 6 numbers from the numbers 1-45. (The order does not matter.) a.) How many unique cards are possible? (One play, or six numbers per card) b.) If a computer could generate 1 card per second, how long would it take to generate all the possible cards? c.) Do you think it would be worthwhile to play every possible card? Why or why not?

Combinations Problems Problem 5: Imagine the same state lottery as in Problem 4 has prizes for matching 3, 4, 5, or 6 out of the 6 chosen numbers. a.) How many unique winning cards are possible? b.) If a player matching 3 numbers wins $5, about how much would you expect a player to win for matching all 6 numbers?

Combinations Problems Problem 6: A pizzeria offers 9 different toppings. A pizza must have a minimum of 1 topping, and can have any combination of toppings up to the maximum of all 9 toppings. Customers also get to choose from among 3 types of crust. How many different pizza varieties are possible?

Combinations Problems Problem 7: A hockey coach is trying to select a starting lineup for the next game. The team has 12 forwards, 6 defensemen, and 2 goalies. The starting lineup will consist of 3 forwards, 2 defensemen, and 1 goalie. How many different groups of 6 players could the coach select to start? (Assume that all 12 forwards are able to play any forward position—left wing, center, or right wing—and all 6 defensemen can play either right or left defense.)

Combinations Problems Problem 8: How many different sums of money could you arrange if you had 1 penny, 1 nickel, 1 dime, and 1 quarter? (Hint: Include all combinations of 1, 2, 3, or 4 coins.)

Combinations Problems Problem 9: A regular deck of cards consists of 4 suits: Hearts and Diamonds are red; Spades and Clubs are black. Each suit has the cards 2-10, as well as a Jack, Queen, King, and Ace. So there are 4 suits of 13 cards each, for a total of 52 cards. a.) How many pairs are possible in a deck of cards? b.) In how many ways could you get 2 pair (e.g., two 4s and two Kings) if you pick 4 cards from a deck?

Combinations Problems Problem 10: A regular deck of cards consists of 4 suits: Hearts and Diamonds are red; Spades and Clubs are black. Each suit has the cards 2-10, as well as a Jack, Queen, King, and Ace. So there are 4 suits of 13 cards each, for a total of 52 cards. In how many ways could you get a flush if you pick 5 cards from a regular deck? (A flush is 5 cards all of the same suit.)