Lesson 10.1 Fundamental Counting Principle and Permutations

1. Lesson 10.1 Fundamental Counting Principle and Permutations

2. Fundamental Counting Principle: You are going on vacation. You pack the following: One pair of khakis, one pair of jeans, one blue t-shirt, one green t-shirt, one striped button down shirt, a pair of tennis shoes and a pair of sandals. How many outfits can you make? List out all of the different outfits you can make in your groups.

3. For a few choices, it’s easy to draw a diagram. As the choice increase, you should probably change the type of diagram you use.

4. At a restaurant, you have a choice of 8 different entrees, 2 different salads, 12 different drinks, and 6 different desserts. How many different dinners consisting of 1 salad, 1 entrée, 1 drink and 1 dessert can you choose?

5. 2 8 12 6 Dessert Salad Drink Entree 1,152 Different Dinners

6. Police use photographs of various facial features to help witnesses identify suspects.

7. One basic identification kit contains 195 hairlines, 99 eyes and eyebrows combinations, 89 noses, 105 mouths, and 74 chins and cheeks combinations. The developer of this kit claims that billions of faces can be produced. How many faces can the kit produce and are the developers claims correct?

8. 105 195 99 89 74 Hairlines Eyes and eyebrows Noses Mouths Chins and Cheeks 13,349,986,650 Different Faces

9. Permutations: Definition: An ordering of a certain number of objects is a permutation of the object. For Example: There are six permutations of the numbers 1, 2 and 3: See if you can list all 6 in your groups: 123 132 213 231 312 321

10. The fundamental counting principle can also be used on permutations. Suppose we have six flower pots to put in a row. How many choices do we have for the first position? The second position? Now fill in the rest and multiply. How many permutations are there?

11. In general, the number of permutations of “n” distinct objects is: n! (n factorial) = n* (n-1) * (n-2) * … * 3 * 2 * 1 There is a factorial function on your graphing calculator ~ Enter the value for n, press MATH, arrow over to PRB, arrow down to number 4 (!), and then press enter.

12. You have homework assignments from 5 different classes to complete this weekend. How many ways can you pick the first 2 assignments to work on? Unfortunately you can’t just do “5!” for this problem since you only are filling in for the first 2 positions.

13. This problem is called a permutation of n things taken r at a time and the formula is: In our problem, n = 5 assignments and r = only 2 assignments being used 5P2 = 5!/(5-2)! = (5 * 4 * 3 * 2 * 1)/( 3 * 2 * 1) = 5 * 4 = 20

14. The general format for a problem like this is: nP2 = n!/(n-r)! n is the total number of items and r is how many you are taking.

15. Permutations with Repetition So far we’ve done permutations of distinct objects – no repetitions. Now let’s look at permutations of duplicate objects.

16. What are the permutations of the letters in DAD? In your groups come up with all of the different combinations – keep in mind the red “D” is different than the black “D” ADD DAD DDA DDA DAD ADD

17. Since each of these sets is identical without the color-coding, there are only 3 distinguishable permutations. So, instead of 3! = 6, our permutation is 3!/2! = 3 The 2! Is because we repeated the D twice.

18. How many distinguishable permutations are there of the letters in OHIO? There are 4 letters, and we repeat the “O” twice, so: 4!/2! = 4 * 3 = 12

19. How many distinguishable permutations are there of the letters in ARKANSAS? ARKANSAS has 8 letters of which A is repeated 3 times and S is repeated 2 times. 56 8!/(3! * 2!) = 40,320/720