Methods of Counting

1. Methods of Counting By Dr. Carol A. Marinas

2. Fundamental Counting Principle • Event M can occur in m ways • Event N can occur in n ways • The event M followed by N is m*n ways IfEvent M is flipping a coin, there are 2 outcomes. If Event N is rolling a die, there are 6 outcomes. So the number of outcomes of flipping a coin and rolling a die is 2 * 6 or 12 ways. Can you list them?

3. Permutations of Unlike Objects How many ways can you choose 2 marbles from the 5 different colored marbles above if order matters? (G, R), (G,B), (G, W), (G, Y) (R, G), (R, B), (R, W), (R, Y) (B, G), (B,R), (B, W), (B, Y) (W, G), (W, R), (W, B), (W, Y) (Y, G), (Y, R), (Y, B), (Y, W)

4. Permutation Formula • An arrangement of things in a definite order with no repetition. • nPr = n ! (n - r)! • 5P2 = 5 ! (5 - 2)! = 5 * 4 ways

5. Combinations of Unlike Objects How many ways can you choose 2 marbles from the 5 different colored marbles above if order does not matter? (G, R),(G,B),(G, W),(G, Y) (R, G),(R, B),(R, W),(R, Y) (B, G),(B,R),(B, W),(B, Y) (W, G),(W, R),(W, B),(W, Y) (Y, G),(Y, R),(Y, B),(Y, W)

6. Combination Formula • An arrangement of things in which order does not matter with no repetition. • nCr = n ! r! (n - r)! • 5C2 = 5 ! 2! (5 - 2)! = (5 * 4)/2 ways or 10 ways

7. Permutation of Like Objects • Before the objects were distinctly different, what if some objects were alike? • How many ways can you arrange the letters in “tot”? tot, tto, ott 3 ways

8. Permutation of Like Objects • How many ways can you arrange the letters in “Mississippi”? • This would be a lot of work to list. There must be an easier way! • Formula n ! r1! r2! r3! … rk! For “tot”, it is 3 ! = 3 ways 2! 1!

9. Permutation of Like Objects • How many ways can you arrange the letters in “Mississippi”? • Formula n ! r1! r2! r3! … rk! For “Mississippi”, it is 11 ! = 34650 ways 1! 4! 4! 2! 11 letters so n = 11 1 M, 4 I’s, 4 S’s, and 2 P’s

10. Thanks for Counting with me!