Methods of Counting By Dr. Carol A. Marinas Fundamental Counting Principle Event M can occur in m...

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Methods of Counting By Dr. Carol A. Marinas

Transcript of Methods of Counting By Dr. Carol A. Marinas Fundamental Counting Principle Event M can occur in m...

Page 1: Methods of Counting By Dr. Carol A. Marinas Fundamental Counting Principle Event M can occur in m ways Event N can occur in n ways The event M followed.

Methods of Counting

By Dr. Carol A. Marinas

Page 2: Methods of Counting By Dr. Carol A. Marinas Fundamental Counting Principle Event M can occur in m ways Event N can occur in n ways The event M followed.

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

If Event 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?

Page 3: Methods of Counting By Dr. Carol A. Marinas Fundamental Counting Principle Event M can occur in m ways Event N can occur in n ways The event M followed.

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)

Page 4: Methods of Counting By Dr. Carol A. Marinas Fundamental Counting Principle Event M can occur in m ways Event N can occur in n ways The event M followed.

Permutation Formula An arrangement of things in a

definite order with no repetition.nPr = n ! (n - r)!

5P2 = 5 ! (5 - 2)! = 5 * 4 ways

Page 5: Methods of Counting By Dr. Carol A. Marinas Fundamental Counting Principle Event M can occur in m ways Event N can occur in n ways The event M followed.

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)

Page 6: Methods of Counting By Dr. Carol A. Marinas Fundamental Counting Principle Event M can occur in m ways Event N can occur in n ways The event M followed.

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

Page 7: Methods of Counting By Dr. Carol A. Marinas Fundamental Counting Principle Event M can occur in m ways Event N can occur in n ways The event M followed.

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

Page 8: Methods of Counting By Dr. Carol A. Marinas Fundamental Counting Principle Event M can occur in m ways Event N can occur in n ways The event M followed.

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!

Page 9: Methods of Counting By Dr. Carol A. Marinas Fundamental Counting Principle Event M can occur in m ways Event N can occur in n ways The event M followed.

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 = 111 M, 4 I’s, 4 S’s, and 2 P’s

Page 10: Methods of Counting By Dr. Carol A. Marinas Fundamental Counting Principle Event M can occur in m ways Event N can occur in n ways The event M followed.

Thanks for Counting with me!


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