Permutations and Combinations Fundamental Counting Principles EQ: What is the difference between a permutation and a combination?

Fundamental Counting Principle In a sequence of n events in which the first one has k1 possibilities, the second has k2 possibilities and the 3rd has k3 and so on, the total number of possibilities of the sequence will be k1 · k2 · k3 ·… ·kn

Example 1 A buffet menu consists of the following selections entrée: chicken, beef, pork, pasta vegetable: corn, string beans, peas potato: baked, french fries, scalloped Find the total number of possible dinner combinations if you can only select one of each item. 4·3·3 = 36

Example 2: A photographer has 5 photographs that she can mount on a page in her portfolio. How many different ways can she arrange her photographs? = 120 5 4 3 2 1

Example 3: The digits 0, 1, 2, 3, and 4 are to be used in a four-digit ID card. How many different cards are possible if repetitions are permitted? if repetitions are not permitted? 5 4 3 2 5 5 5 5 b) = 120 a) = 54 = 625

Example 4: Using the digits 1, 2, 3, and 5, how many 4 digit numbers can be formed if The 1st digit must be a 1 and you can repeat digits? The 1st digit must be a 1 and you can’t repeat digits? The number must be divisible by 2 and you can repeat digits? The 1st number must be prime and you can’t repeat digits.

Example 4: Using the digits 1, 2, 3, and 5, how many 4 digit numbers can be formed if 1 4 4 4 1 3 2 1 a) = 43 = 64 b) = 6 4 4 4 1 3 3 2 1 c) = 64 d) = 18

Example 5: A father, mother, 2 boys, and 3 girls are asked to line up for a photograph. Determine the number of ways they can line up if… there are no restrictions on how they stand. the parents must stand together. the parents do not stand together. all females must stand together.

Example 5: 7 6 5 4 3 2 1 a) ______ ______ ______ 6 1 = 5040 6 1 b) ______ ______ ______ = 1440 (2X1) 5 4 3 2 c) part a – part b = 5040-1440= 3600 d) ______ ______ ______ = 576 (4x3x2x1) 4 3 2 1

PERMUTATIONS An arrangement of ‘n’ objects in a specific order. 3 basic types: The permutation of all ‘n’ objects is n! n! = n(n-1)(n-2)….·1

Example: Permutation Rule 1 How many ways can 9 trophies be displayed on a shelf? 9! = 362,880

Permutation Rule 2 Use when you need to arrange only ‘r’ of the original ‘n’ objects.

Example: Permutation Rule 2 10 ladies are up for Miss Universe. How many different ways can they be ranked? 10! = 3,628,800 Of the ten ladies, how many ways can they pick the top 3 contestants? = 720

Permutation Rule 3 To be used when you need to arrange ‘n’ objects of which k1, k2,…etc are alike.

Example: Permutation Rule 3 Find the number of ways you can arrange the letters of the following names: Tyler 5! = 120 Shannon 7!/3! = 840 Jennifer 8!/(2!2!) = 10080

Summarizing Video on Permutations

COMBINATIONS A selection of ‘r’ objects from ‘n’ objects without regard for order.

Example 1: Combinations A University catalog lists 15 graduate courses in physics. Calculate the number of ways in which a student can select 4 courses to take. = 1365

Example 2: Combinations A travel brochure lists 10 museums in the city of London. In how many ways can a tourist visit 4 museums if The order in which the museums are visited does not matter The order in which the museums are visited does matter?

Example 2: Combinations - Answers Since the order does not matter, this part of the problem is a combination Since the order does matter, this part of the problem is a permutation

Example 3: Combinations A student committee must consist of 3 juniors and 4 seniors. If seven juniors and eight seniors are willing to serve on the committee, in how many different ways can the committee be formed? = 35 · 70 = 2450

Example 4: Combinations Tyler wants chocolate ice cream in a dish with 2 different toppings. How many different combinations are possible? His Pappy wants 3 different toppings on any one flavor of ice cream. How many different combinations are possible? Flavors Toppings Vanilla Hot fudge Chocolate Sprinkles Peanut butter Peanuts M & M’s

Summarizing Video on Combinations