Permutations and Combinations

Questions:

What are permutations? Combinations? In what types of situations would you apply each one?

Launch:

Your family is ordering an extra-large pizza. There are four toppings to choose from (pepperoni, sausage, bacon, and ham). You have a coupon for a three-topping pizza.

1.) Determine all the different three-topping pizzas you could order. You may want to create a list, diagram, table, or chart to show possible outcomes and counting techniques.

2.) Think about the pizza topping combinations you found in the previous lesson. You chose three toppings from four. Determine how many ways you can assemble a pizza with ONLY three toppings (pepperoni, sausage, bacon). This will depend on the order that ingredients are placed on the pizza. For example, putting on pepperoni, then sausage, then bacon is different than putting on bacon, then pepperoni, then sausage. Show how you determined your list.

Investigation:(Adapted from www.omegamath.com)

Example 1:

Suppose you work at a music store and have four CDs you wish to arrange from left to right on a display shelf. The four CDs are hip-hop, country, rock, and alternative (shorthand: H, C, R, A). How many options do you have?

Solution: If you select H first then you still have three options remaining. If you then pick C, you have two CDs to choose from. You can find the number of ways to arrange your display by the factorial rule: for the first choice (event) you have 4 choices; for the second, 3; for the third, 2; and for the last, only 1. The total ways then to select the four CDs are: 4! = (4)(3)(2)(1) = 24.

Factorial Rule: For n items, there are n! (pronounced n factorial) ways to arrange them.

n! = (n)(n - 1)(n - 2). . . (3)(2)(1)

For example: 3! = (3)(2)(1) = 6 4! = (4)(3)(2)(1) = 24 5! = (5)(4)(3)(2)(1) = 120 6! = (6)(5)(4)(3)(2)(1) = 720 Note: 0!=1

Try solving this problem:

How many ways can six different radio commercials be played during a one-hour radio program?

Permutation – A permutation is used to describe a counting procedure in which order matters.

If order matters, then AB is not the same as BA.

Consider the following:

If you have three friends to send text messages to, Taylor, Justin, and Aubrey, how many different ways can you text your friends if order matters?

Therefore TJA represents texting Taylor first, then Justin, and then Aubrey.

Because order matters, another way to text your friends is TAJ.

All of the possible arrangements are:

TJA, TAJ, JTA, JAT, ATJ, AJT

OR, 3! = (3)(2)(1) = 6 ways

Try solving this problem:

If you have four friends to send text messages to, Taylor, Justin, Arsenio, and Aubrey, how many different ways can you text your friends if order matters?

Summary

”If you want to arrange n objects in groups of n at a time, there are ___! ways to accomplish this task.

Property: There are _________ ways to arrange n objects in groups of n at a time.”

Now, let’s say you have four friends, but only need to text three of them when order matters. In order to find the number of arrangements you must use the permutation formula.

Permutation Formula

n = the total number of items you have from which to chooser = the number you are actually going to use

Can r be larger than n? Explain your answer.

Can they be equal? Explain your answer.

Example 2:

Let’s say you have four friends, but only need to text three of them when order matters. Find the number of ways to text your friends.

Solution:

There are 24 ways to test three out of your four friends if order matters.

Try this problem:

How many different ways can a city building inspector visit five out of six buildings in the city if she visits them in a specific order?

Combination - A combination is a counting situation in which order is not important.

If order does not matter, then AB is the same as BA.

Example 3:

The art club has 4 members. They want to choose a group of three to compete in a regional competition. How many ways can three members be chosen?

Because the members of the group do not have specific jobs, order does not matter. In other words, choosing Taylor, Aubrey, and Justin in the same as choosing Aubrey, Taylor, and Justin. Therefore, we need to use the combination formula to solve this problem.

 Combination Formula

n = the total number of items you have from which to chooser = the number you are actually going to use

Solution:

There are 4 ways to chose 3 people for the competition when order is not important .

Try this problem:

How many ways are there to select three bracelets from a box of ten bracelets disregarding the order of the selection?

Example 4:

How many ways can three cars and four trucks be selected from eight cars and eleven trucks to be tested for a safety inspection?

1. Would you use permutation or combinations? How do you know?

2. First consider the cars. How many ways are there to select three cars from eight cars?

3. Next consider the trucks. How many ways are there to four trucks for from eleven trucks?

4. Now consider the full question:How many ways can three cars and four trucks be selected from eight cars and eleven trucks to be tested for a safety inspection?

In a previous lesson, you learned that “and” means to ____________.

Final answer: ____________

Conclusions:

Based on what you have learned in the investigation, answer the following questions:

How would you determine when to use permutations? Combinations?

In Class Problems:

1. How many different ways can the letters in the word "store" be arranged? Show how you know.

2. How many ways can 10 people be placed in alphabetical order according to their first names? Show how you know.

3. A club with 20 women and 15 men needs to form a committee of size 7.

a. How many committees are possible? Explain how you know.

b. How many committees are possible if the committee must have 4 women and 3 men? Show how you know.

c. How many committees are possible if the committee must have at least 1 man? Explain how you know.

d. How many committees are possible if the committee must have 7 executive members (assigned positions)? What is different about this problem compared to a, b, and c.

4. How many ways are there to deal a five-card hand consisting of three eight's and two sevens? Tell why you chose permutations or combinations. Show how you know.

5. A certain marathon had 50 people running for first prize, second, and third prize.

A)   How many different possible outcomes are there for the first three runners to cross the finish line?

B)   How many ways are there to correctly guess the first, second, and third place winners? How do you know?

Homework:For each question (a) explain why you chose permutations or combinations and (b) show how you determined your solution.

1. The general manager of a fast-food restaurant chain must select six restaurants from eleven for a promotional program. How many possible ways can this selection be done?

2. How many ways can a committee of four people be selected from a group of ten people?

3. How many different four letter permutations can be formed from the letters in the word decagon?

4. How many ways can seven floral arrangements be arranged in a row on a single display shelf?

5. How many ways can a baseball manager arrange a batting order of nine players?

6. How many ways can four baseball players and three basketball players be selected from twelve baseball players and nine basketball players?

7. If a person can select three presents from ten presents under a Christmas tree, how many different combinations are there?

8. In a board of directors composed of eight people, how many ways can a chief executive officer, a director, and a treasurer be selected?

Resources

www.omegamath.com/Data/d2.2.html

Question Standards SummaryLaunch1 MM1P1b b. Solve problems that arise in mathematics and in other

contexts. MM1P5a a. Create and use representations to organize, record, and

communicate mathematical ideas. 2 MM1P1b b. Solve problems that arise in mathematics and in other

contexts. MM1P3d d. Use the language of mathematics to express mathematical

ideas precisely. MM1P5a,c a. Create and use representations to organize, record, and

communicate mathematical ideas. c. Use representations to model and interpret physical, social, and mathematical phenomena.

Investigation1-4 MM1P1a-c a. Build new mathematical knowledge through problem solving.

b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems.

MM1D1b b. Calculate and use simple permutations and combinations. Summary MM1P2b b. Make and investigate mathematical conjecture. 5a MM1P1b,d b. Solve problems that arise in mathematics and in other

contexts. d. Monitor and reflect on the process of mathematical problem solving.

MM1P3a,b,d a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. d. Use the language of mathematics to express mathematical ideas precisely.

5b,c MM1P1a-c a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems.

MM1D1b b. Calculate and use simple permutations and combinations.5d MM1P1a-c a. Build new mathematical knowledge through problem solving.

b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems.

MM1D1a a. Apply the addition and multiplication principles of counting. Conclusions MM1P1a-d a. Build new mathematical knowledge through problem solving.

b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving.

MM1P3a,b,d a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. d. Use the language of mathematics to express mathematical ideas precisely.

In Class 1-5 MM1P1b-d b. Solve problems that arise in mathematics and in other

contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving.

MM1P3a,b,d a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. d. Use the language of mathematics to express mathematical ideas precisely.

1,2,3b,c,4 MM1D1a,b a. Apply the addition and multiplication principles of counting. b. Calculate and use simple permutations and combinations.

3a,d,5 MM1D1b b. Calculate and use simple permutations and combinations.Homework1-8 MM1P1b-d b. Solve problems that arise in mathematics and in other

contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving.

MM1P3a,b,d a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. d. Use the language of mathematics to express mathematical ideas precisely.

6 MM1D1a,b a. Apply the addition and multiplication principles of counting. b. Calculate and use simple permutations and combinations.

1-4, 7,8 MM1D1b b. Calculate and use simple permutations and combinations.