Permutations and Combinations

1
Permutations and Combinations

• Independent Events
• Events that do not affect each other
• Spinning a number 6 and then spinning a number 5
  on the same spinner.
• Picking an ace out of a deck of cards and rolling
  a di and getting a 6
• Picking an ace of hearts out of a deck of cards,
  then picking a jack of clubs out of the same deck
  with replacement.
• Dependent Event
• Events that affect each other.
• Picking an ace of hearts out of a deck of cards,
  then picking a jack of clubs out of the same deck
  without replacement.

2
Permutations and Combinations

• Basic Counting Principle
• When one event can be chosen p ways and another
  event can be chosen q ways, then the events can
  have pq ways
• Example How many ways can Mr. Vogle make an ugly
  outfit if he has 5 pairs of ugly pants, 6 ugly
  shirts and 2 pairs of shoes?
• 56260

3
Permutations and Combinations

• Permutations
• P(n,n)n! (permutation of n things taken n at a
  time)
• P(n,r) n!/(n-r)!
• Order matters
• Example How many ways can you arrange 5 subject
  books?
• P(5,5)
• 120

4
Permutations and Combinations

• Combinations(order is not important)

5
Permutations and Combinations

• How many ways can you order 4 different class
  officers from a class council of 12 people?
  (Does order matter)
• P(12,4)
• How many ways can you order a 4 person sub
  committee from a class council of 12 people?
• C(12,4)

6
Permutations and Combinations

• At Kennedy High School there are 12 names on the
  ballot for sophomore class officers. Six will be
  selected to form a class committee.
• a. How many different committees of 6 can be
  formed?
• b. In how many ways can a committee of 6 be
  formed if each student has a different
  responsibility?
• c. If there are 7 girls and 5 boys on the ballot,
  how many committees of 3 girls and 3 boys can be
  formed?
• Answer to A
• A. Order is not important in this situation, so
  the selection is a combination of 12 people
  chosen 6 at a time.C(12, 6)924

7
Permutations and Combinations

• At Kennedy High School there are 12 names on the
  ballot for sophomore class officers. Six will be
  selected to form a class committee.
• b. In how many ways can a committee of 6 be
  formed if each student has a different
  responsibility?
• Answer
• P(12, 6) 665,280

8
Permutations and Combinations

• At Kennedy High School there are 12 names on the
  ballot for sophomore class officers. Six will be
  selected to form a class committee.
• c. If there are 7 girls and 5 boys on the ballot,
  how many committees of 3 girls and 3 boys can be
  formed?
• Answer
• Since the events are independent, the answer is
  the product of the combinations C(5, 3) and C(7,
  3).
• C(5, 3) ? C(7, 3) 350