1. Day 2 - Combinations

2. Warm-up
Tyler must choose a 4 digit password for his bank account. Find the number of possible 4 digit passwords using 4 different digits.
Bill, Allie, James, and Cindy are friends. In how many different ways can they stand in a row for a photo?
How many items are in the sample space of flipping a coin once?
Steve must choose four characters for his computer password. Each character can be any letter from A through Z or any digit from 0through 9. All letters and digits may be used more than once. Find the number of possible passwords.
How many permutations in GEOMETRY?

3. Combinations ORDER DOES NOT MATTER!
A Combination is an arrangement of items in which order does not matter.
ORDER DOES NOT MATTER!
Since the order does not matter in combinations, there are fewer combinations than permutations.  The combinations are a "subset" of the permutations.

4. Combinations
To find the number of Combinations of n items chosen r at a time, you can use the formula

5. Combinations
To find the number of Combinations of n items chosen r at a time, you can use the formula

6. Combinations
To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible?
Practice:

7. Example:
A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions?

8. Multiple Events:
When finding the number of ways both an event A and event B can occur, you need to _________________
When finding the number of ways that an event A or event B can occur, you __________.

9. Example:
Parents have 10 books that they can read to their children this week. Five of the books are nonfiction and 5 are fiction.
A. If the order in which they read the books is not important, how many different set of 4 books can they choose?
B. How many groups of 4 books are there in which all of the books are either nonfiction or fiction?

10. The Student Senate consists of 6 seniors, 5 juniors, 4 sophomores, and 3 freshmen.
How many different committees of exactly 2 seniors and 2 juniors can be chosen?
How many different committees of at most 4 students can be chosen?

11. Example: multiple events
A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards?
120 ways

12. Example:
A restaurant has 6 different kinds of hamburgers on the menu and 5 different kinds of soda. How many different ways can someone select 2 different hamburgers and a soda for lunch?

13. Example:
Anthony is going to college this year. When he went to sign up for his classes he had a choice of 4 math classes and 5 computer classes. He must sign up for one math class and 2 computer classes. How many ways can he do this?
40 ways

14. A class has 10 boys and 20 girls in it. Find the
number of ways that the teacher could select a
team of 3 students to work on a project if the team
must consist of:
a) Any 3 students b) 3 girls   
c) 1 boy and 2 girls d) at least 2 girls
1140 ways
4060 ways
3040 ways
1900 ways

15. Permutation or Comb. Practice:
A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated?
Answer Now

16. Permutations or Comb.? Practice:
From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled?
Answer Now

17. Permutations or Combinations??
Practice:
To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible?
Answer Now

18. Determine whether the following will use a Permutation or a Combination:
 a. Your school newspaper has an editor in chief and an assistant editor in chief. The staff of the newspaper has 12 students. In how many ways can students be chosen for these 2 positions?
b. Thirty people apply for 10 job openings as welders. How many different groups of people can be hired?
c. In how many ways can 4 distinct positions for a relay race be assigned from a team of nine runners?
D. In how many ways can a club with 15 members elect a president, vice president, secretary and treasurer?

19. Evaluate the following expressions: