1. Combinations

2. Combinations Objectives: (1) Students will be able to use combinations to find all possible arrangements involving a limited number of choices. Essential Questions: (1) What are combinations and how can we find them?

3. Combinations What is a Combination? - Have you ever played a sport and thought about how many different ways the coach could have assigned players to the starting lineup? - A COMBINATION is an arrangement or listing in which order IS NOT important.

4. Combinations How Do I Find The Value of A Combination? - We calculate the value of a combination in the following way: C(5,3) =

5. Combinations How Do I Find The Value of A Combination? - We calculate the value of a combination in the following way: 5 x 4 x 3 60 5 x 4 x 3 60 3 x 2 x 1 6 3 x 2 x 1 6 Start with this number Count down this many numbers C(5,3) = = = 10

6. Combinations Real World Example: If Coach Bob McKillop has 12 basketball players on his team, how many ways can he choose 5 players to start a game?

7. Combinations Real World Example: If Coach Bob McKillop has 12 basketball players on his team, how many ways can he choose 5 players to start a game? P(12,5) 12 x 11 x 10 x 9 x 8 P(12,5) 12 x 11 x 10 x 9 x 8 5! 5 x 4 x 3 x 2 x 1 5! 5 x 4 x 3 x 2 x 195040 120 120 C(12,5) = = = = 792 ways

8. Combinations Example 1: Combinations. Find the value of C(6,3).

9. Combinations Example 1: Combinations. Find the value of C(6,3). P(6,3) 6 x 5 x 4 120 P(6,3) 6 x 5 x 4 120 3! 3 x 2 x 1 6 3! 3 x 2 x 1 6 C(6,3) = = = = 20

10. Combinations Example 2: Combinations. Find the value of C(15,2).

11. Combinations Example 2: Combinations. Find the value of C(15,2). P(15,2) 15 x 14 210 2! 2 x 1 2 2! 2 x 1 2 C(15,2) = = = = 105

12. Combinations Example 3: Election Candidates. How many ways can a delegation of 4 people be selected from a class of 22 students?

13. Combinations Example 3: Election Candidates. How many ways can a delegation of 4 people be selected from a class of 22 students? P(22,4) 22 x 21 x 20 x 19 P(22,4) 22 x 21 x 20 x 19 4! 4 x 3 x 2 x 1 4! 4 x 3 x 2 x 1 175,560 175,560 24 24 C(22,4) = = = = = 7315 ways

14. Combinations Example 4: Birthday Party. Sommer is having a birthday party. She has narrowed the list to 9 people, but she can only take 4. How many combinations of friends are possible?

15. Combinations Example 4: Birthday Party. Sommer is having a birthday party. She has narrowed the list to 9 people, but she can only take 4. How many combinations of friends are possible? P(9,4) 9 x 8 x 7 x 6 3024 P(9,4) 9 x 8 x 7 x 6 3024 4! 4 x 3 x 2 x 1 24 4! 4 x 3 x 2 x 1 24 C(9,4) = = = = 126 ways

16. Combinations Real World Example: Taste Test. A taste test of 9 different soft drinks is held at Ferndale. If each taster is randomly given 5 of the drinks to taste, how many combinations of soft drinks are possible?

17. Combinations Real World Example: Taste Test. A taste test of 9 different soft drinks is held at Ferndale. If each taster is randomly given 5 of the drinks to taste, how many combinations of soft drinks are possible? P(9,5) 9 x 8 x 7 x 6 x 5 15120 P(9,5) 9 x 8 x 7 x 6 x 5 15120 5! 5 x 4 x 3 x 2 x 1 120 5! 5 x 4 x 3 x 2 x 1 120 C(9,5) = = = = 126 ways

18. Combinations Guided Practice: Find the value. (1) C(8,3) = ? (2) How many three card hands can be dealt from a deck of 52 cards?

19. Combinations Guided Practice: Find the value. (1) C(8,3) = 56 (2) How many three card hands can be dealt from a deck of 52 cards? 22,100 different ways

20. Combinations Independent Practice: Find the value. (1) C(6,4) = ? (2) How many ways can you choose four items from a Chinese menu of 14 items?

21. Combinations Independent Practice: Find the value. (1) C(6,4) = 15 (2) How many ways can you choose four items from a Chinese menu of 14 items? 1001 different ways

22. Combinations Real World Example: Quiz Questions. On a English quiz you are allowed to answer 4 out of the 6 six questions. How many ways can you choose the questions?

23. Combinations Real World Example: Quiz Questions. On a English quiz you are allowed to answer 4 out of the 6 six questions. How many ways can you choose the questions? P(6,4) 6 x 5 x 4 x 3 360 P(6,4) 6 x 5 x 4 x 3 360 4! 4 x 3 x 2 x 1 24 4! 4 x 3 x 2 x 1 24 C(6,4) = = = = 15 ways

24. Combinations Summary: - Combinations involve arrangements or listings where order is not important. - We use the following notation: C(9,4) = C(9,4) = * The symbol C(9,4) represents the number of combinations of 9 possible things to take, and we are taking 4 of them

25. Combinations Summary: - Combinations involve arrangements or listings where order is not important. - We use the following notation: C(9,4) = = Start with this number Count down this many numbers Combination 9 x 8 x 7 x 6 4! 4 x 3 x 2 x 1 = 126

26. Homework: Combinations