Counting principle

Identify the steps, count the choices per step, multiply across. The single most-used move in the topic. License plates, PINs, meals, passwords.

01 Mini-Lesson 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 What This Looks Like on ALEKS

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

License plates (3 letters + 3 digits) v1

Each license plate in a certain state has six characters (with repeats allowed). The first three are letters of the alphabet; the last three are digits 0 through 9. How many license plates are possible in this state?

4-digit PIN v2

A bank PIN consists of 4 digits, each chosen from 0 through 9 with repeats allowed. How many distinct PINs are possible?

Restaurant meal combinations v3

A diner offers 5 appetizers, 8 entrees, and 4 desserts. A complete meal is one of each. How many different meals are possible?

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 How to Solve It

total = n₁ × n₂ × ··· × nₖ
multiply the choices at each independent step

When a process has multiple independent steps, the total number of outcomes is the product across all of them.

Identify the steps.

Six slots, one character per slot. Letters in slots 1-3; digits in slots 4-6. All slots are independent (repeats allowed).

Count choices per slot.

slots 1-3 (letters): 26 each slots 4-6 (digits): 10 each

Multiply across.

26³ \times 10³ = 17,576 \times 1,000 = 17,576,000

About 17.5 million possible plates — more than any state will ever issue.

▸ COMMON SLIPS(1) Added instead of multiplied. 26 + 26 + 26 + 10 + 10 + 10 = 108 is what you get with addition. The principle multiplies. (2) Forgot the "repeats allowed" language. When repeats aren't allowed (no two slots same letter), the choices shrink at each step (26, 25, 24, ...) and you're in permutation territory (Q4). (3) Missed a slot. Count the characters carefully. Off-by-one in slot count gives an answer that's off by a factor of 26 or 10.

04 Practice — Step by Step

Practice on a different multi-step count.

4-digit PIN.

A 4-digit PIN, each digit 0-9 with repeats allowed. How many possible PINs?

total =

Restaurant meal.

5 appetizers, 8 entrees, 4 desserts — one of each. How many meals?

meals =

▸ NICE WORK

You've walked through the whole problem.

That's the move. ALEKS will give you a different version with different numbers — but the steps are the same.

Q1 Q3