# Combinations (Make sets, combining one from each group)

Finding simple combinations when merging two sets of events looks tricky, but itâ€™s not. **As long as you are required to use one of each item,** all you have to do is multiply all of the different options in one set times all of the different options of another set.

Why does this work?

**In order to see how combinations work, it's often help to make a tree. A tree allows you to see all of the different combinations, and count them. Make enough trees and you'll be ready to trust the multiplication!**

*Example: *You have two shirts (pink and blue) and three hats (black, brown, and red), that you love. If you combine them into outfits, how many different possible outfits do you have:

As you can see, each shirt can be matched with each hat for a total of 6 outfits. The shortcut here is to multiply 2 (shirts) times 3 (hats) for a total of 6 outfits.

This process (drawing the trees) and this shortcut (multiplying) work for more complicated combination problems as well:

*Example: *At the cafeteria there are 3 different sandwiches students can choose for lunch (tuna, turkey, peanut butter) and three different desserts they can choose (pie, cake, ice cream). There are 4 different drink options (soda, lemonade, water, milk). If you want one sandwich, one dessert, and one drink, how many lunch combinations are possible?

First, figure out which items or events needs to be combined:

- 3 different sandwiches
- 3 different desserts
- 4 different drink options

Next, multiply the number of options in each set by the number of options in each other set (because, each time you get a different sandwich, you could combine that with any of the 3 desserts, and any of the 4 drinks!).

$3\times3\times4=36 \text{ possible meals}$

Combination problems in which you combine one of each option are easy: multiple the options together and you will get the total number of combinations possible!

#### Practice Problems:

## Combinations (Make sets combining one from each group)

Find how many combinations you can make:

- You are rolling a six-sided die one time and flipping a coin one time. How many different combinations can you get?
- You are rolling a six-sided die two times and flipping a coin twice. How many different combinations can you get?
- You packed for a two day trip and brought two each of shorts, shirts, socks, underwear, and hats. On the first day, how many different outfits might you put together from these items?
- You are at an ice cream shop with 20 flavors of ice cream and two types of cone. How many different cones might you put together?
- If, at the same ice cream shop, you are choosing two flavors of ice cream and a type a cone, how many different cones might you put together?
- If you are wrapping a gift and can choose amoung 5 types of wrapping paper and 3 types of ribbon, how many differently wrapped packages can you make?
- If you are choosing a special card to go with that gift and can choose from among 6 cards, 3 different envelopes, and 2 types of confetti (or no confetti at all), how many different cards can you make?
- You are putting together a summer schedule. You are choosing between a math or a physics class, a horsebackriding or cooking camp, and either yoga, pilates, or karate competitions. If you choose one of each, how many different summer schedules can you make?