# Combinatorics for Breakfast

The Hendrix Flying Squirrels and Sugar Gliders are the Ultimate Frisbee teams of Hendrix College in Conway, Arkansas. They came to Fayetteville, home of FreshPlans, for a tournament, and they fueled up for the day’s games with a good breakfast.

This gave us an excellent opportunity to practice combinatorics.

Here’s what was on the table:

The menu included two egg dishes: quiche and strata; two breads: muffins and biscuits; and three fruits: oranges, kiwi fruits, and apples. Click to embiggen the picture.

That’s a total of seven different items to choose from at breakfast time. If each player chose just one item, they’d have 7 choices.

If all the Hendrix players chose two items, they’d have all these choices:

1. quiche and muffins
2. quiche and biscuits
3. quiche and apples
4. quiche and kiwi fruit
5. quiche and oranges
6. quiche and strata
7. strata and muffins
8. strata and biscuits
9. strata and apples
10. strata and kiwi fruit
11. strata and oranges
12. muffins and biscuits
13. muffins and apples
14. muffins and kiwi fruit
15. muffins and oranges
16. biscuits and apples
17. biscuits and kiwi fruit
18. biscuits and oranges

That’s 18 more possibilities.

If each player chose 1 egg dish, 1 kind of bread, and 1 fruit, they’d have these possibilities:

1. quiche and muffins and apples
2. quiche and muffins and kiwi fruit
3. quiche and muffins and oranges
4. quiche and biscuits and apples
5. quiche and biscuits and kiwi fruit
6. quiche and biscuits and oranges
7. strata and muffins and apples
8. strata and muffins and kiwi fruit
9. strata and muffins and oranges
10. strata and biscuits and apples
11. strata and biscuits and kiwi fruit
12. strata and biscuits and oranges

That’s 12 more options. But really, some players might want strata and quiche and apples (as in the photo below) or quiche and  kiwi fruit and oranges or muffins, oranges, and kiwi fruit.

Since there are 7 things, there are only 7 choices for the first item. If everyone has just one thing for breakfast, there are 7 possibilities.  If everyone chooses a second item, they have seven choices again. It’s 7 times 7, or 49 possibilities. But a lot of those are duplicates. Once we remove the duplicates, we have 18, as we saw in the list above. When everyone chooses a third item, if we don’t remove duplicates, we have 343 different possible breakfasts. Lots of those have duplicates.

Is this getting confusing?

Let’s just look at the fruits first. There are three kinds of fruit: kiwi fruit, oranges, and apples.

For the first choice, there are three possibilities:

1. kiwi fruit
2. oranges
3. apples

For the second choice, each of the three possibilities has three options.

1. kiwi fruit
1. plus oranges
2. plus apples
3. plus more kiwi fruit
2. oranges
1. plus more oranges
2. plus apples
3. plus kiwi fruit
3. apples
1. plus oranges
2. plus more apples
3. plus kiwi fruit

We therefore end up with 9 options: 3 for the first choice x 3 for the second choice.

If we allow duplication, then we’ll always have the number of first choices times the number of second choices, and so on.

For the breakfast, with one egg dish, one kind of bread,and one kind of fruit, we’d have

(2 egg dishes x 2 kinds of bread= 4) x 3 fruits= 12

Present this information to your class, using manipulatives or sentence strips, and make sure the basic concept is clear. You can watch this video about the Flying Squirrels’ lunch options to reinforce the idea:

Then divide into small groups and have students make some more calculations:

• All the Flying Squirrels wear black shorts, but they can choose between dark orange or light orange shirts. How many possible outfits can they wear to play their sport? (1 pants  x 2 shirts= 2)
• The Flying Squirrels played at the university and the Sugar Gliders played at a school field. After the game, they chose between going to a restaurant for dinner or going to their host families for dinner. After dinner, there was a party, but some of the players chose to stay home and play games. How many choices of activities did each student have? (1 field, since the students didn’t have a choice, x 2 dinners x 2 after dinner activities= 4)

Once the students have the concept, have each small group prepare a combinatorics problem for the other groups to figure out. Allow the use of manipulatives or drawings. Have each group choose one problem they’ve solved and plan a way to present it to the rest of the class with visual aids or movement.