Transcript Combinations Problems

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 10question 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.)