Combinations (Make sets, combining one from each group)

Finding simple combinations when merging two sets of events looks tricky, but it’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!