Lesson 3

Lesson 3: Combinations

 

Focus

 

In his book The Codebreakers: The Story of Secret Writing, David Kahn tells the story of how a New York City detective broke up a betting ring. The detective observed the bookie taking bets and writing them down. However, when the detective made the arrest, much to his surprise, the list did not consist of names and betting amounts. Rather, the list contained a series of musical notes.

 

Convinced that the notes must be a code, the detective took the list for cryptanalysis. Sure enough, when played, the notes did not result in beautiful music. The notes were a code! When broken, each code provided the identity of the person and the amount of the bet. The detective solved his case.

 

There have been other cases of music notes being used to deliver hidden messages. The code might be in the lyrics or within the notes themselves.

The piano is one of the most common musical instruments in the world. A typical keyboard has 88 keys with a repetitive naming sequence as shown in the following partial keyboard diagram. Keys with the same name, such as two A notes, sound similar, but one note is higher than the other.

A chord can be produced on a piano when more than one key is pressed at the same time. How can you determine the number of different chords a piano can produce? What if you want chords using only the white keys? Suppose the pianist’s fingers could only span eight white keys. How can you determine how many one-handed chords can be produced?

 

In Lesson 2 you examined permutations—the number of ways objects can be arranged when order is important. On the piano, the order in which a chord is written or played does not change the sound. ACE is the same as CAE since all keys are pressed at the same time in a chord.

 

 

In this lesson you will explore a method of determining the number of ways to select objects if the order is not important.

 

Lesson Outcomes

 

At the end of this lesson you will be able to determine the number of combinations of elements given certain criteria.

 

Lesson Questions

 

In this lesson you will investigate the following questions:

• What are the similarities and differences between permutation and combination problems?
• How are problems involving combinations solved?
• How do you solve equations involving combinations?

Assessment

 

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 3 Assignment (Download the Lesson 3 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Glossary Terms and Formula Sheet
• work under Project Connection

Discover

 

Try This 1

 

Suppose you are planning to open an ice cream stand and have 4 kinds of ice cream: chocolate, vanilla, strawberry, and tiger. You plan to serve 2 different scoops at a time. Use Two Scoops to determine how many two-flavour cones can be made.

 

1. If you wanted to advertise the number of different two-flavour cones available, would you calculate the number available by using 4P2 = 12? Explain.
2. If you added another flavour, blueberry, how many different cones could you produce?

Save your responses in your course folder.

 

Share 1

 

With a partner or group, discuss the following questions based on what you discovered in Try This 1.

1. In Two Scoops, what do you suppose the 4 and the 2 represent in 4C2 if the C stands for combination?
2. Think of another scenario where you might use combinations instead of permutations.

 If required, save a record of your discussion in your course folder.

Explore

 

In Try This 1 you looked at permutations and combinations of ice cream cones. The number of combinations is the number of ways you can select items from a group when order does not matter. In Two Scoops, a chocolate-strawberry cone is the same as a strawberry-chocolate cone.

 

You may have noticed in Try This 1 that the notation for combinations is similar to that of permutations.

 

	Notation	Order	Ice Cream Cone Possibilities
Combination	nCr	Order doesn’t matter.	4C2 = 6
Permutation	nPr	Order matters.	4P2 = 12

 

In Try This 1, nCr represents taking r objects (two flavours) from a group of n objects (four possible flavours), when order does not matter. In nPr, the restriction on r is r ≤ n.

 

This restriction on r is also true for nCr because you can’t choose more objects than are available.

 

In Try This 2 you will explore combinations and compare them to permutations.

 

Try This 2

 

Step 1: Open “Permutations and Combinations.”

 

In “Permutations and Combinations” you see the number of ways you can take items from a box. You are also allowed to decide whether or not to pay attention to the order in which you take the items from the box.

 

Step 2: Set “Number of tiles in box” to 3. Set “Number of draws from box” to 2, and then select “Yes” to indicate the order is important, as shown in the diagram. This setting corresponds to the first line of the following table. Hit “Simulate.” Switch between the tabs at the top for different representations of the information.

 

Screenshot reprinted with permission of ExploreLearning

 

Complete a table like the one shown by changing the tiles, draws, and order of importance.

 

Tiles	Draws	Is order important?	Number of Possibilities	Sketch of Tree Diagram	Notation	Permutation or Combination
3	2	yes	6		3P2	permutation
3	2	no
5	2	yes		Do not draw.
5	2	no		Do not draw.
4	3	no		Do not draw.
4	4	no		Do not draw.
5	5	no		Do not draw.

 

Save your responses in your course folder.

 

Share 2

 

With a partner or group, discuss the following questions based on what you discovered in Try This 2.

1. How did the tree diagram change when you switched from order being important to being not important?

2. Why are there fewer combinations of a specified number of items as compared to the number of permutations?
3. “Permutations and Combinations” shows that the permutation and combination formulas are related by Explain why dividing the number of permutations by r! gives the number of combinations.
4. Think of a real-life example where you would be interested in the number of combinations rather than the number of permutations.

If required, save a record of your discussion in your course folder.

In “Permutations and Combinations” you may have noticed that you could represent the number of combinations 5C3 as The number of combinations, nCr, is equal to the number of permutations, nPr, divided by the number of ways the elements could be arranged, r!

 

In Two Scoops, you saw 2!, or 2, duplicates for each cone, so there were twice as many permutations as combinations.

 

Your calculator likely has a nCr button that will calculate the number of combinations for you. Note that some textbooks use an alternate notation for combinations where
In Two Scoops you could have written and known that this meant 4C2. You evaluated this equation and got 6.

 

The combination function on your calculator will likely be found in the same menu as the permutation function. Make sure you can evaluate combinations and permutations using your calculator. Check with your teacher if you have trouble finding these functions.

 

Self-Check 1

 

A Grade 12 class of 120 students is forming a graduation committee.

1. In how many ways can a committee of 10 be chosen? Use two notations to represent your answer. Answer
2. What is the total number of 10-person committees that can be formed? Answer
3. Explain why 120P10 is divided by 10! in the formula to find 120C10. Answer

So far you have seen that can be used to determine the number of combinations. It is also possible to rewrite this formula without the permutation component.

 

 

So, the combination formula is as follows:

 

As with permutations, there are often extra conditions placed on the choices to be made. If, in Self-Check 2, the coach had to make choices based on positions, the number of combinations would change. Consider the scenario in Try This 3.

 

Try This 3

 

A hockey coach wishes to choose a first line. The coach has to choose 1 of 5 centres, 1 of 7 right wingers, 1 of 6 left wingers, and 2 of 10 defence players. In how many ways can the coach choose the first line?

 

Use a table similar to the following table to help solve the problem.

 

Position	Centre	Right Wing	Left Wing	Defence
Number of Players to Choose From	5	7	6	10
Number on Ice at One Time	1	1	1	2
Combination Notation
Number of Ways to Fill Each Position
Total Number of Ways					total = ?

 

 

Hemera/Thinkstock

 

Save your responses in your course folder.

 

Share 3

 

With a partner or group, discuss the following questions based on your answer to Try This 3.

1. Compare the ways your group determined how the starting line could be formed.
2. Of the methods you saw, describe the one that seems most efficient.

If required, save a record of your discussion in your course folder.

In Try This 3 the coach must choose a centre and a right wing and a left wing and two defence. By using the fundamental counting principle, you multiply the number of choices for each position to determine the total number of ways to make these choices.

• centre: 5C1 
• right wing: 7C1
• left wing: 6C1 
• two defense: 10C2

The total number of ways the coach can choose the first line is 5 × 7 × 6 × 45 = 9450 ways.

 

Hemera/Thinkstock

 

You used cases in some of the permutation calculations in Lesson 2. You would add the number of permutations in each case to determine the total. There are also some combination questions that require the use of cases.

 

Pianists can create different chords by pressing several keys down at the same time on a piano.

 

Think about how many chords are possible in each of the following situations.

 

• Using the keys from the piano as shown, how many different chords—combinations of two or more keys—can be produced using exactly four fingers if each finger only presses one key, black or white, at a time?
• How many chords can be produced using three or four fingers?
• How many chords can be produced using two or three or four fingers?

Watch Piano Chords to see how the number of chords can be calculated for each situation.

 

In Piano Chords you saw that cases were required because you could use two, three, or four fingers. The number of combinations for each case was then added to determine the total number of combinations.

Self-Check 5

Connect

 

Lesson 3 Assignment

Lesson 3 Summary

 

Combinations can be used to solve counting problems where the order is not important. A combination can be represented by nCr or where n is the size of the group from which the objects are taken and r is the number of objects that are taken at a time. The following formula can be used to calculate a combination where 0 ≤ n ≤ r:

 

 

Many problems will require you to calculate multiple combinations and then apply the fundamental counting principle.