Counting principle

Count the options in each row, then multiply across. The only real decision is multiply rather than add, and the numbers themselves tell you which.

01 Worked Example Video

▸ VIDEO COMING SOON

A short walkthrough explaining what you need to know and how to solve this question type lands here once it's recorded.

02 Recognize the Question

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

Three categories

A sandwich comes with a choice of bread (White, Rye, Sourdough), a filling (Cheese, Ham), and a side (Crisps, Salad, Soup, Slaw).

How many different sandwiches are possible?

With a yes/no row

A pizza order chooses a size (Regular, Large), a crust (Thin, Thick, Stuffed, Gluten-free, Sourdough), a topping (Pepperoni, Mushroom, Pepper), and whether or not to add extra cheese.

How many different orders are possible?

Spot the slip

A bike is ordered with a frame size (4 options), a colour (3 options) and a handlebar type (3 options). A student answered 10.

What is the correct number of bikes?

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 The Rule

total = n₁ × n₂ × n₃ × … one factor per independent choice

every combination of choices is a different item which is why they multiply

Each choice is made independently of the others: picking a large size does not change which buns are available. So every one of the size options pairs with every bun option, and the counts multiply rather than add.

Common slips

(1) Adding instead of multiplying. The single most common error, and the answer it gives is far too small to be plausible. (2) Miscounting a row. Commas in the possibilities column are what you are counting; read them carefully. (3) Skipping a yes/no row. “With cheese or without” is two options and belongs in the product.

04 Practice — Step by Step

Count each row, multiply across

When ordering a drink at Fern & Filter, the customer chooses a size, a milk, a syrup, a number of espresso shots, and a lid type.

Here are the possibilities for each choice.

Choice	Possibilities
Size	Small, Medium, Large
Milk	Whole, Oat, Almond, None
Syrup	Vanilla, Caramel, Hazelnut, Mocha, None
Shots	Single, Double
Lid	Sipper, Flat

How many different drinks are possible?

Step 1

Count the rows

Write the five counts in order, separated by commas. Size first.

Step 2

Multiply or add?

How do the five counts combine?

Step 3

First two rows

3 sizes × 4 milks.

Step 4

All five

12 × 5 × 2 × 2.

Q17 Q19

Counting principle

240 against 16