Tennis Balls Meeting
(Counting)
Topic
This meeting’s topic is counting, combinations (order does not matter) and permutations (order does matter).
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Materials Needed
♦ 2007-2008 MATHCOUNTS tennis ball poster
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Alex needs three tennis balls.
From the yellow, pink and white balls on this court, how many distinct color combinations can Alex select?
Visit www.mathcounts.org/poster for the solution to this problem.
Three cans of three tennis balls from different manufacturers or with different numbers printed on them
- Optional (other distinguishable items could certainly be
1420 King Street, Alexandria, VA 22314 703-299-9006, info@mathcounts.org www.mathcounts.org
Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation © 2007 MATHCOUNTS Foundation used)
Ice cream party essentials with three different flavors of ice cream - Optional
The printing of this poster was made possible by
NATIONAL SPONSORS:
Lockheed Martin, Raytheon Company, Northrop Grumman
Foundation, Texas Instruments Incorporated, National
Society of Professional Engineers, 3M Foundation, General
Motors Foundation, CNA Foundation, ConocoPhillips,
National Council of Teachers of Mathematics, National
Aeronautics and Space Administration
Meeting Plan
The task is to answer the question on the poster. However, before sending kids off to solve this problem, be sure the students understand what is being asked.
Do we have to include all three colors?
( No.)
Does the order in which we select the balls matter?
( No, choosing a white and then two pinks is the same as choosing two pinks and then a white. The word “combinations” is a math term that implies that the order does not matter.)
Have students work on the problem.
How did they get to their answers? If they made an organized list, how were they sure that they did not leave out any options? ( 10 combinations: YYY, PPP, WWW, WWP, WPP, WWY, WYY,
PPY, PYY, YWP )
You can have students work out this problem with actual balls. You probably do not have white, pink and yellow balls, but you could get three cans of balls from three different manufacturers, or better yet, three cans of balls from the same manufacturer, but with different numbers on them.
(Every ball in a can of balls will have the same number.) This will enable students to “act out” the scenario.
What if Alex is going to put the balls into a tennis ball can that holds three balls? How many distinct cans of balls can he create?
Ask your kids to explain in other words what we want them to do now. Do they understand that the order now matters? A white ball at the top with two pink balls below it will look different than a can with the white ball on the bottom and two pink balls above it.
Your students’ organized lists now might take the form of a tree diagram. If so, that’s great. If not, listen to their solutions, praise the good ideas – even if the good idea did not lead to the right answer – and then show them a tree diagram for the situation. They will see that there are
27 different possibilities for the cans. Ask the students how this list of 27 can be used to get the answer of 10 for the original poster problem. Which final outcomes of the tree diagram are the same when the order of the colors does not matter?
2007-2008 MATHCOUNTS Club Resource Guide 21

After they have all drawn the tree diagram for the situation (or you have drawn a nice big one that they can all see), start improvising with your questions and let them be creative. There are many great extensions from this activity. After you ask a few questions, see if any of the students can come up with extension questions.
Sample extension questions:
If there are 10 pink, 10 white and 10 yellow balls, what is the probability that Alex will pick the same color for all three balls? What is the probability that Alex will select three different colors?
What if there were 10 balls of a fourth color added to the court? How many color combinations are possible when selecting three balls? Or if the three balls will be put into a can, how many distinct cans are possible?
If there is exactly one purple ball on the court in addition to the pink, yellow and white ones shown, how many color combinations are possible if Alex is selecting three balls?
Possible Next Steps
What is another situation in which a tree diagram can be used to count the options?
Once students come up with a few of their own, here’s one they might really like… They need to take three scoops of ice cream, and there are three different flavors of ice cream available. If
Aaron is putting the three scoops into a bowl, the order doesn’t matter. If he is putting them on a cone, the order (top, middle, bottom) can matter. This example provides a great excuse to bring ice cream to the club meeting!! And including toppings just adds to the math discussion.
Please share your photos, ideas and success stories with us by e-mailing them to info@ mathcounts.org with the subject line “MATHCOUNTS Club Program.” We’d love to see you and your students at your ice cream party!
22 2007-2008 MATHCOUNTS Club Resource Guide

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Visit www.mathcounts.org/poster for the solution to this problem.
1420 King Street, alexandria, Va 22314 703-299-9006, info@mathcounts.org www.mathcounts.org
Founding Sponsors: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Foundation © 2007 MATHCOUNTS Foundation
The printing of this poster was made possible by
NatioNal SpoNSorS: