14.2 - The Multiplication Principle

Links and useful resources

Concept summary and lesson

Combinations of options - multiply to find how many.

We're continuing to count things today. This time, we're going to count the number of ways that we can configure a bunch of options. First, let's get some terms straight:

An option is one thing you could choose that's part of a group of interchangeable things (one shirt out of all of the shirts)
A setting is a group of interchangeable options that you have to decide on in order to answer the question. (settings would be which shirt, which pants, which socks, which shoes if all of those things have some options you can decide between)

I just made those terms up, they're not part of "the lexicon" so to speak. But, here's how they work:

Whenever you're trying to figure out how many ways something could turn out, you can go through it a setting at a time and figure out how many options are left at that point for that setting. If all of them are independent, then you just multiply the number of options available at each setting together and that's your answer!

Examples:

Make a one-topping pizza. You have three sauces, two cheeses, and seven toppings.
- Once you pick a sauce (three options), you can pick any cheese. So for each of those three sauces, you have two pizzas ( $3 \times 2 = 6)$
- Once you pick a cheese, you can put any of seven toppings on the pizza. Since there are $6$ pizzas currently possible, each one of them could turn out in $7$ different ways, so you get $6 \times 7 = 42$ possible one-topping pizzas
Now make a two-topping pizza, where the toppings are different.
- We still have $42$ from the one-topping pizza question. Now we need to add another topping, but how many options are there for this setting?
- Since they have to be different, there are only $6$ toppings left that we could choose.
- So, for each of our $42$ possible pizzas, we could add one of $6$ new toppings, leading to $42 \times 6 = 252$ possible pizzas
How many ways can you assign three jobs to people in a 6-person patrol? Nobody can do more than one job.
- This time, we have to fill three spaces, so that means three settings.
- The first setting has 6 options because there are 6 people without jobs.
- The second has 5, because after we picked the first one there will be five people left. But, the 5 people left will be different for each option from the first selection, so there are $5 \times 6 = 30$ ways to pick the first two
- Finally, we have to pick the third person, but now there are only $4$ people left for options. So, we end up with $6 \times 5 \times 4 = 120$ ways to assign the tasks.

The whole trick of counting in this way is to figure out what the settings are, and how many options each one has. If you can think of a way to give the settings names, it makes it easier to think about the problem. In the patrols/jobs problem, we called the settings "positions we need to fill," which made it easy to see that there are three of them and each one will have a different number of people for options.

Guided practice

How many ways can I get dressed if I have 3 shirts, 5 pairs of pants, and 2 pairs of shoes?
In how many ways can I arrange four different books from left to right on a shelf?
Your math club has 16 members. In how many ways can it select a president, a vice-president, and a treasurer if no member can hold more than one office?
In how many ways can we form an international commission if we must choose one European country from among 6 European countries, one Asian country from among 4, one North American country from among 3, and one African country from among 7?

Homework

#hw (math) (gr7) 14.2 The Multiplication Principle (p. 519) 2025-04-14