The fundamental counting principle.
When a process has multiple independent steps, the total number of outcomes is the product of the number of choices at each step. The single most-used move in the topic.
Imagine asking: "How many different license plates can a state issue if every plate is three letters followed by three digits?" Writing them all out would take roughly a year of round-the-clock typing. Listing the sample space directly is hopeless.
But the problem has hidden structure: a plate is built one slot at a time. The first slot has 26 choices, the second slot has 26, the third has 26, the fourth has 10, the fifth has 10, the sixth has 10. The total number of plates is simply the product: 26 × 26 × 26 × 10 × 10 × 10 = 17,576,000.
That move — "multiply the choices at each step" — is the fundamental counting principle, and it's the workhorse of every counting question in the topic. License plates, PIN codes, dinner menu combinations, three-medal podium orderings, lottery tickets — same move, different numbers.
Multiply the choices at each step.
total = n₁ × n₂ × n₃ × ··· × nₖ
"Independent" here means the choices at one step don't restrict the choices at another. Picking the first letter of a license plate doesn't stop you from picking any letter as the second — all 26 are still available. When the steps DO affect each other (no repeats allowed), you'll need permutations or combinations from Lesson 4.
Six slots, two groups: three letter slots at 26 choices each and three digit slots at 10 choices each. The product across all six gives 17,576,000 possible plates — one for every combination the state could ever issue.
The fundamental counting principle states that for any multi-step procedure where the steps are independent, the total number of possible outcomes equals the product of the number of choices available at each step.
Words you'll see on ALEKS
- Independent steps Steps where the choice at one step does not constrain the choices at any other step. License-plate slots are independent (letter A in slot 1 doesn't block letter A in slot 2); medals in a race are not (the gold medalist can't also win silver).
- With replacement / repeats allowed After making a selection at step k, that choice is still available at step k+1. License plates allow repeats. A bag where you draw a ball, record it, and put it back is "with replacement."
- Without replacement / no repeats Once a choice is made, it can't be made again. Medals (gold, silver, bronze) can't go to the same person twice. The number of choices shrinks at each step (n, n−1, n−2, ...). That setup is a permutation, introduced in L4.
- Tree diagram A visual organizer for multi-step problems with few choices. Each step adds a layer of branches; the total leaves at the bottom equals the product of the branches per step. Useful for sample-space problems with 2-3 steps and small choice counts.
- Shrinking choices In a multi-step count where repeats aren't allowed, the number of choices drops by 1 at each step. "Pick 3 books from 10 with no repeats" = 10 × 9 × 8. The pattern of descending integers is the seed of the factorial.
Counting license plates.
"Each license plate in a certain state has six characters (with repeats allowed). The first three characters are letters of the alphabet; the last three are digits 0-9. How many license plates are possible in this state?"
Identify the steps.
Six slots to fill, one character per slot. Each slot is an independent step (repeats allowed). So six independent steps.
Count the choices per step.
Each of the first three slots: 26 letters in the alphabet. Each of the last three slots: 10 digits (0 through 9).
slots 4-6: 10 each
Apply the principle: multiply.
Total plates = 26 × 26 × 26 × 10 × 10 × 10. Compute in two halves:
10³ = 1,000
total = 17,576 × 1,000 = 17,576,000
Just over 17.5 million possible plates, far more than any single state will ever issue.
→ multiply across the slotsThree multi-step counts.
A bank PIN is 4 digits long. Each digit can be anything from 0 to 9, with repeats allowed. How many possible PINs are there?
A diner offers 5 appetizers, 8 entrees, and 4 desserts. If a complete meal is one of each, how many different meals are possible?
A 7-digit phone number (within a single area code) is being assigned. The first digit must be 2 through 9 (not 0 or 1). The remaining six digits can be anything 0-9. How many such phone numbers are possible?
Three fast questions before you move on.
Q1. Two coins are flipped. How many possible outcomes are there?
Q2. An outfit consists of one shirt and one pair of pants. There are 7 shirts and 3 pairs of pants. How many outfits?
Q3. When does the fundamental counting principle not apply in its simple form?
When you can't list, you count.
The fundamental counting principle is the answer to "how many possible outcomes?" whenever the sample space is too big to write out. License plates, PIN codes, passwords, lottery tickets, restaurant menu combinations, gene sequences — the math is the same in every case: identify the steps, count the choices at each, multiply.
It's also the engine that powers L4 (permutations and combinations) when the steps stop being independent: arranging three medalists out of ten runners uses 10 × 9 × 8 = 720, a shrinking-choices version of the same multiply move.
Next: Lesson 3 turns to multi-event probability — what happens when you want P(A or B), and what the "overlap" correction handles when A and B can both happen at once.
Continue to Lesson 03Different angle? Need another rep? These are optional — tap any that look helpful.
The Fundamental Counting Principle
Multiple worked examples (clothing combinations, license plates) walking through the multiplication-across-stages logic. Tight, exam-ready treatment.
Fundamental Counting Principle, Tree Diagrams, and Probability
Builds tree diagrams alongside the multiplication rule — the visual companion that Khan and OCT don't lean into as heavily. A short reach outside the preferred list to cover the tree-diagram bonus.