PERMUTATIONS AND COMBINATIONS M408 – Probability Unit

Permutation – number of ways to arrange certain items in a distinct order.

Combination- # of ways to choose a group of items from a larger set (ORDER DOES NOT MATTER).

Example 1 – Given 4 letters, {A,B,C,D} 2-Letter Permutations2-Letter Combinations AB, AC, AD, BC, BD, CD BA, CA, DA, CB, DB, DC AB, AC, AD, BC, BD, CD 12 unique arrangements of 2 letters 6 groups of two letters (order doesn’t matter)

Example 2 -

Example 3 -  A.) There are 10 competitors in the dance-off. How many ways could you award 1 st, 2 nd, and 3 rd place?  B.) 10 people try out for a dancing role in a music video. Three people will be chosen. How many different groups of 3 could be chosen?

Example 4 -  How many ways are there to line up 8 students?

Example 5 -  There are 12 people in your ghost-hunting club. How many ways are there to select a 4-person exploratory committee to venture into the spooky basement?

Calculator Time

Standard Deck of Cards 52 total cards:26 black26 red 4 suits:clubsspadesheartsdiamonds kinds: Ace, 2, 3,…, 10, Jack, Queen, King (face cards)

Example 6 -  A.) How many 5-card hands are possible in poker?  B.) How many 5-card hands with 2 hearts and 3 clubs?  C.) 5-card hands with 2 Jacks and 2 Kings?  D.) 5-card hands with 4 of a kind?  E.) 5-card hands with at least 3 face cards?

When picking for specific ‘sub-groups’ from your total…  Use a Combination for each subgroup, as well as one for the ‘leftovers’.  Multiply the Combinations together (Slot Method) (Ex. 6B,C,D)  “At Least/Less Than/At Most/More Than”: Consider each distinct scenario, Add the Combinations. (Ex. 6E)

Example 7 -  If 22 players try out for the baseball squad, and you select 15 members, how many different ways are there to pick the team?  Once the 15 players have been selected, how many ways are there to make a batting order for the first match?

Example 8 - Your 12 member club consists of 5 men and 7 women. Find the number of ways to pick a.) A 3-person party-planning committee b.) A president, vice-president, and secretary c.) A 3 person committee with 2 women and 1 man.

Permutation or Combination?  Words that suggest Permutation (order matters). Arrange, Order, Line up, Distinct Positions  Situations to use Combination (order doesn’t matter). Pick a group (no order or position specified), Playing Cards

Example 9 – Permutation or Combination? a.) You want to pick a 4 person committee out of 12 people. b.) You want to pick a 7-card hand from a deck of 52 cards. c.) From 15 club members select 3 different officer positions. d.) Pick 3 letters from the word HYPOTENUSE. e.) Arrange 3 letters from the word HYPOTENUSE. f.) You want to order 5 people in a line. g.) You have 5 lawn seats to a concert, and 12 friends. h.) For which type of problem could you sometimes use slot method? (C or P)

Example 10 – find n.