Date of the Problem
December 26, 2022

We can use the Fundamental Counting Principle to determine the number of combinations of the various options for each gingerbread cookie. The first option is which of the 4 colors to use for the eyes. Then there are 3 options for the nose and only 1 option for the mouth. Next, the gum drop buttons can be one of 4 different flavors. Finally, each cookie has either a green hat or a yellow hat. That means there are 4 × 3 × 1 × 4 × 2 = 96 different combinations of these characteristics. Therefore, the number of gingerbread cookies Santiago baked was 96/12 = 8 dozens.

Suppose that instead of choosing one of the four different gum drop flavors for the buttons, Santiago decided that each gingerbread child would have three buttons, each made from a different flavored gum drop. If there are four different flavors of gum drops, how many different combinations of three gum drop buttons are possible?

Since there are four different flavors of gum drops from which to choose the three buttons, there are 4C3 = 4!/(3!1!) = 4 different combinations of three buttons that Santiago could possibly choose. Another way to think about the number of combinations of 3 of the 4 different flavored gum drops is that there are 4 ways to select the 1 gum drop flavor that is not used for one of the three buttons.

How many different arrangements of three buttons can Santiago make using the combinations of three different flavored gum drops from the previous problem?

Each of the 4 combinations of three different flavored gum drops from the previous problem can be arranged in 3! = 6 different ways. Therefore, the number of arrangements of three different flavored gum drops that Santiago can make is 4 × 6 = 24 arrangements. Another way to think about the number of arrangements of three different gum drops for the buttons is that there are 4 options for the first button, 3 for the second button and 2 for the third button. Using the Fundamental Counting Principle, that’s a total of 4 × 3 × 2 = 24 arrangements.

Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.

Page 2 contains ONLY PROBLEMS. ♦

Math topic
CCSS (Common Core State Standard)
Difficulty