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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 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.

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methods of counting

Methods of Counting

By Dr. Carol A. Marinas

fundamental counting principle
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?

permutations of unlike objects
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)

permutation formula
Permutation Formula
  • An arrangement of things in a definite order with no repetition.
  • nPr = n !

(n - r)!

  • 5P2 = 5 !

(5 - 2)!

= 5 * 4 ways

combinations of unlike objects
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)

combination formula
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

permutation of like objects
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

permutation of like objects1
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!

permutation of like objects2
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