The difference between permutations and combinations Name Recap

The difference between permutations and combinations Name: ________

Recap A lunch special includes one main item, one side, and one drink. How many different meals can you choose if you pick one main item, one side, and one drink? 4 x 3 =36

Permutations and Combinations A permutation is an arrangement of outcomes in which the order does matter. A combination is a grouping of outcomes in which the order does not matter.

Permutations and Combinations • Note: In order to complete a question on permutations and combinations always ask yourself the following question: • Does the order of the terms matter?

Permutations and Arrangements • A permutation is an arrangement of outcomes in which the order matters. • Questions on permutations and arrangements will usually ask you ‘how many different ways’ can you arrange something. These questions are very specific and will give you a concrete order in which to follow.

ABCDE If I wanted to arrange these letters, how many ways could I do it if there are to be no restrictions ? 5 x 4 x 3 x 2 x 1 = 120 5 options for the 1 st letter 4 options for the 2 nd letter 3 options for the 3 rd letter 2 options for the 4 th letter 1 option for the 5 th letter

• For the more straightforward questions use the box method as shown below: • Question • How many ways can we arrange the letters of the word Ireland if it must begin with the letter E?

Question • In how many ways can the letters of the word marketing be arranged if: 1. There are no restrictions 2. The 3 vowels must be together

• All 3 vowels are not together? • Not together = 1 - (that they are together)

Question Four girls and 5 boys are to be lined up: How many arrangements are possible if: 1. There are no restrictions 2. If 2 particular girls must be together 3. If 2 particular girls must not be together

Permutations and arrangements which involve taking a certain number of objects at a time. • For questions of this format we must utilise the following formula • In the formula n represents the original number of arrangements and r tells us how many are taken at a time. • Question: In how many ways can 7 books be arranged on a shelf taking 4 at a time

Combinations • A combination is a grouping of outcomes in which the order does not matter. • Questions on combinations will usually give you a scenario where the words ‘choose’ or ‘select’ are used. • When answering questions on combinations we must use the n. Cr button on our calculators

Question • In how many ways can a team of 5 players be chosen from 9 players? (Note: we can see here that the order does not matter)

• When we have 2 events in a combination question simply do out each event individually and multiply your answers together • Question Find the number of ways in which a panel of four men and three women can be chosen from seven men and five women.

Question • In how may ways can a committee of 6 be formed from 5 teachers and 8 students if there are to be more teachers than students on each committee. (Hint: There must be at least one student on each committee ) • Note: It is good practice for these questions to list out the possible outcomes in English

The difference between Permutations and Combinations • Here a few examples of combinations (order doesn’t matter) and permutations (order matters). 1. Combination: Picking a team of 3 people from a group of 10. 2. Permutation: Picking a President, VP and Waterboy from a group of 10. A. Combination: Choosing 3 desserts from a menu of 10. C(10, 3) = 120. B. Permutation: Listing your 3 favourite desserts, in order, from a menu of 10. P(10, 3) = 720.

Distinguishing between Combinations and Permutations Tell whether the following situations involve combinations or permutations. Then give the number of possible outcomes. Remember to ask yourself: Does the order of the terms matter?

Question 1 An English test contains five different essay questions labeled A, B, C, D, and E. You are supposed to choose 2 to answer. How many different ways are there to do this?

Question 3 A family of 3 plans to sit in the same row at a movie theater. How many ways can the family be seated in 3 seats?

Question 4 Ingrid is stringing 3 different types of beads on a bracelet. How many ways can she string the next three beads if they must include one bead of each type?

Question 5 John wants to order a sandwich with two of the following ingredients: mushroom, eggplant, tomato, and avocado. How many different sandwiches can John choose?

Question 6 A group of 8 swimmers are swimming in a race. Prizes are given for first, second, and third place. How many different outcomes can there be?

Question 7 How many different ways can 9 people line up for a picture?

Question 8 Four people need to be selected from a class of 15 to help clean up the campus. How many different ways can the 4 people be chosen?

Question 9 Four people need to be selected from a class of 15 to help clean up the campus. How many different ways can the 4 people be chosen, if the only two girls refuse to help?

Question 10 A basketball team has 12 members who can play any position. How many different ways can the coach choose 5 starting players?

Question 11 A basketball team has 12 members who can play any position. How many different ways can the coach choose 5 starting players if the captain MUST play the first half?

Question 12 When ordering a pizza, you can choose 2 toppings from the following: mushrooms, olives, pepperoni, pineapple, and sausage. How many different types of pizza can you order?

Question 13 Nine people in a writing contest are competing for first, second and third prize. How many ways can the 3 people be chosen?

Question 14 You are ordering a triple-scoop ice-cream cone. There are 18 flavours to choose from and you don’t care which flavor is on the top, middle, or bottom. How many different ways can you select a triple-scoop ice-cream cone?