Counting Strategies for Probability – Designing Color Schemes
Carlos and Marya own and operate an interior design company. You would like to hire them to assist you in
redecorating a children’s playroom in your house. To help you get started, Carlos and Marya bring a picture and 6
coordinating paint chips to give you ideas. Originally, you thought that you would paint the room with just one
color, but after seeing the picture, you realize that you could use several colors in the playroom. Since this is to be a
playroom, your preference is to use bright colors, such as red, blue, yellow, green, orange, and/or purple. Knowing
your color preferences, Carlos and Marya have brought 6 bright colors of paint chips for you to choose from.
1. Use the paint chips to lay the 6 available colors out on your table.
(a) Illustrate the total number of colors available below.
All Available Colors
(b) Choose your favorite color from among these colors. (Each person in your group may choose a different favorite
color, if desired.) Circle your favorite color in the picture above.
(c) If you only want one color for your playroom and your significant other is in charge of choosing one of these
colors and he/she chooses randomly, what is the probability that she/he chooses your favorite color? (Think about
the definition of probability. The sample space for this event is illustrated above!)
2. Look at the 6 available colors. If you want to use two colors in your playroom, there are several choices. Carlos
and Marya brought multiple copies of each paint chip color. Use the paint chips to lay out all possible pairs of
colors.
(a) Illustrate the total number of two-color choices available.
All Available Two-Color Choices
(b) Look for the two-color choices that contain your favorite color. Circle each of these in the picture above.
(c) If your significant other is in charge of choosing two colors for your playroom and he/she chooses randomly,
what is the probability that he/she makes a two-color choice that includes your favorite color? (Think about the
definition of probability. The sample space for this event is illustrated above!)
3. As you can see, even with a limited number of colors available, there are many color combinations possible for
redecorating a room. The number of such color combinations depends on how many colors are available and how
many colors you desire. Carlos and Marya have a chart that helps them determine the number of color combinations
that clients can choose when they bring a certain number of sample paint colors. Each time, the client can reject all
of the colors, accept all of the colors, or accept some subset of the colors.
How many three-color combinations are possible if five colors are available? How can these combinations be
“built” from the two-color combinations and three-color combinations in the row above?
Number of Color Combinations Possible (with Illustrations)
0 colors
1 color
2 colors
3 colors
chosen
chosen
chosen
chosen
1
1
1 color
available
4 colors
chosen
2 colors
available
1
2
1
3 colors
available
1
3
3
1
4 colors
available
1
4
6
4
1
5 colors
available
1
5
10
10
5
5 colors
chosen
1
4. Here is another version of Carlos and Marya’s chart, this time without illustrations. Look for a pattern from one
row to the next and use this pattern to fill in the remaining three rows.
What does the chart tell you about the number of two-color combinations possible if six colors are available? Does
this match your information in Exercise 2?
Number of Color Combinations Possible
0
1
2
3
colors
color
colors colors
chosen chosen chosen chosen
1
1
1 color
available
1
2
1
2 colors
available
1
3
3
1
3 colors
available
1
4
6
4
4 colors
available
1
5
10
10
5 colors
available
6 colors
available
7 colors
available
8 colors
available
4
colors
chosen
5
colors
chosen
6
colors
chosen
7
colors
chosen
8
colors
chosen
1
5
1
5. Given that Carlos and Marya have brought 6 sample paint colors to show you, many choices are available to you,
since you haven’t yet decided how many colors you wish to use in painting the playroom. You can reject all the
colors, choose one color, choose two colors, choose three colors, choose 4 colors, choose 5 colors, or accept all 6
colors. Use the chart above to calculate how many such choices you have.
If your significant other is in charge of choosing a color combination, and he/she just chooses randomly, what is the
probability that he/she chooses
(a) a three-color combination?
(b) a two-color combination or a three-color combination?
(c) a color combination with 4 or more colors?
6. Let’s suppose that your playroom needs new paint on the ceiling and on the walls and you have chosen a twocolor combination, say blue and yellow, from the 6 available colors. Now, you need to decide which color to paint
the ceiling and which color to paint the walls. Illustrate the possible ways that you can paint the playroom.
ceiling
ceiling
walls and do not knowwalls
If the painters arrive at your house
which color goes where, what is the probability that they paint
the ceiling yellow and the walls blue?
7. Let’s suppose that you add molding to the playroom and you have chosen a three-color combination, say blue,
yellow, and red, from the 6 available colors. Now you need to decide which color to paint the ceiling, which color
to paint the walls, and which color to paint the molding. We need to find a way to systematically illustrate the
possible ways that you can paint the playroom. One way to do this is with a tree diagram. A tree diagram allows us
to “build” all possible three-color assignments, one color at a time. In the first column of branches, illustrate the
possible ceiling colors. In the second column, illustrate the possible ceiling-wall color assignments. In the third
column, illustrate the possible ceiling-wall-molding color assignments.
Ceiling Colors
Ceiling and Wall
Colors
If the painters arrive and do not know which colors go where,
(a) what is the probability that they paint the ceiling red?
(b) what is the probability that they paint the walls yellow?
(c) what is the probability that the molding is not blue?
Ceiling, Wall, and
Molding Colors
8. Let’s suppose your playroom has paneling and molding that need paint, in addition to the ceiling and the walls.
You have chosen a four-color combination, say blue, yellow, red, and purple, from the 6 available colors. Now you
need to decide which color to put where. We could systematically illustrate the possible ways that you can paint the
playroom, using a tree diagram, as we did in the three-color case, but as you can see from the partial tree diagram
below, this is a little unwieldy. Based on what the complete tree diagram should look like, find a way to calculate
the number of possible four-color assignments, without having to illustrate all possible such assignments in a tree
diagram.
Ceiling Colors
Ceiling and Wall
Colors
Ceiling, Wall, and
Molding Colors
Ceiling, Wall, Molding,
and Paneling Colors
If the painters arrive and do not know which colors go where,
(a) what is the probability that they paint the ceiling red?
(b) what is the probability that they paint the walls yellow and the ceiling red?
(c) what is the probability that the molding is neither blue nor purple?
Follow-Up Exercises
1. Carlos and Marya’s pictures and 6 coordinating paint chips have inspired you to redecorate another room in your
house, in a similar style to that shown in the picture. Six shades of the color are available.
(a) How many 3-shade combinations can be chosen?
(b) How many 3-shade combinations contain the darkest shade?
(c) If shades are chosen randomly, what is the probability that a 3-shade combination contains the darkest
shade?
2. Eight students are training for the Putnam mathematics competition, including Alicia.
(a) How many 4-person teams can be chosen to compete?
(b) How many 4-person teams including Alicia can be chosen?
(c) If the competition team is chosen randomly, what is the probability that Alicia will be on the team?
3. Ten people try out for the school basketball team, including Michelle and her best friend Romy.
(a) How many 5-person teams can be chosen?
(b) How many 5-person teams including Michelle and Romy can be chosen?
(c) If players are chosen randomly, what is the probability that both Romy and Michelle will make the
team?
(d) Once a 5-person team is selected, in how many ways can the five people be assigned positions?
(e) Assuming the 5-person team includes Michelle and Romy, how many team assignments include
Michelle being assigned the position of “Center”?
(f) If Michelle and Romy both made the team and positions are assigned randomly, what is the probability
that Michelle was assigned the position of “Center”?
4. Fifteen people are running for the school board, including Jesus, Marta, and Nancy.
(a) How many 6-person school boards can be elected?
(b) How many 6-person school boards, including Jesus, Marta, and Nancy can be elected?
(c) If voters simply vote randomly, what is the probability that Jesus, Marta, and Nancy are elected to the
school board?
(d) Once a 6-person board is selected, in how many ways can a chairperson and vice-chairman be selected?
(e) Assuming the 6-person board includes Jesus, Marta, and Nancy, how many chairperson and vicechairmen selections would include at least one of Jesus, Marta, and Nancy?
(g) If Jesus, Marta, and Nancy all were elected to the board and the positions of chairperson and vicechairperson were selected randomly, what is the probability that at least one of Jesus, Marta, and Nancy
were selected for these two positions?
Hint: Many of these situations are similar to “choosing paint colors”. Extend the chart you created for #4 in the
activity to help you solve some of the counting problems above.