Module 8: Permutations, Combinations, and the Binomial Theorem

 

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.”

 

 
This is a play button for Permutations and Combinations.
Screenshot reprinted with permission of ExploreLearning



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.

 

This diagram shows the Permutations and Combinations gizmo with the top right tabs circled and labelled “Tabs to change representation.”

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.    

 

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