Math
Fundamental Counting Principle Calculator
Enter how many choices there are at each stage (say 3 shirts, 4 pants, 2 pairs of shoes) and the calculator multiplies them to count every possible outcome, writing the product out in full.
The second mode handles codes and arrangements: choose k positions from n symbols, either with repetition allowed (a 4-digit PIN: 10^4) or without (a 4-letter code with no repeats: 26 × 25 × 24 × 23), and explains why the two differ.
One positive whole number per stage, separated by commas or spaces. Up to 30 stages.
Try an example
Result
Total outcomes
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- Product
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- Stages
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- In scientific notation
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- Note
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Study path
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What the fundamental counting principle calculator solves
The fundamental counting principle (also called the multiplication rule) says that if a task happens in stages and each stage has a fixed number of options no matter what was chosen before, the total number of outcomes is the product of the options. With 3 shirts, 4 pairs of pants, and 2 pairs of shoes there are 3 × 4 × 2 = 24 outfits. A tree diagram draws every branch; the principle counts the leaves without drawing them.
| Situation | Product | Outcomes |
|---|---|---|
| 3 shirts, 4 pants, 2 shoes | 3 × 4 × 2 | 24 |
| 4-digit PIN | 10 × 10 × 10 × 10 | 10,000 |
| 4-digit PIN, no repeated digit | 10 × 9 × 8 × 7 | 5,040 |
| License plate: 3 letters then 3 digits | 26^3 × 10^3 | 17,576,000 |
| Coin flipped 5 times | 2^5 | 32 |
| Arranging 6 books on a shelf | 6 × 5 × 4 × 3 × 2 × 1 | 720 |
With repetition vs without
When a symbol may be reused, every position has the same number of options, and the count is n^k: 10^4 = 10,000 PINs. When each symbol may be used only once, the options shrink by one at each position: 10 × 9 × 8 × 7 = 5,040. That falling product is the permutation P(n, k) = n!/(n - k)!, so the counting principle is the reason the permutation formula works. If the order of the chosen symbols does not matter, divide by k! to get combinations instead.
How to use it
- For a multi-stage choice, list the number of options at each stage and read the product.
- For codes and arrangements, enter n options and k positions, then say whether repetition is allowed.
- Open Show the work to see the factors, the running product, and the permutation formula when repetition is off.
How to read the answer
The total counts every distinct sequence of choices, one per branch of the tree diagram. Large totals are also shown in Writing a number as one digit, a decimal part, and a power of ten.. Without repetition the count is a permutation and drops to 0 when k exceeds n, because the symbols run out. A stage with 1 option leaves the total unchanged.
Common mistakes and edge cases
- Adding the choices instead of multiplying: 3 shirts and 4 pants give 12 outfits, not 7.
- Ignoring 'no repeats' and using n^k when the options shrink at each stage.
- Using the counting principle when the order of selection does not matter; that needs combinations, which divide out the k! orderings.
- Forgetting restricted stages like a first digit that cannot be 0 or a letter that must be a vowel.
- Counting a forced stage (1 option) as 0 options; it multiplies by 1, not 0.
Worked examples
Outfits
3 shirts × 4 pants × 2 shoes = 24.
Total outcomes
24
License plates
26, 26, 26, 10, 10, 10 → 17,576,000.
Total outcomes
17,576,000
Single stage
Just 5 choices.
Total outcomes
5
4-digit PIN with repetition
10^4 = 10,000.
Total outcomes
10,000
4-digit PIN, no repeats
10 × 9 × 8 × 7 = 5,040.
Total outcomes
5,040
Arrange all 6 books
6! = 720.
Total outcomes
720
Too many positions without repetition
5 positions from 3 symbols: impossible.
Total outcomes
Error
Bad input
Zero or negative stages are rejected.
Total outcomes
Error
Frequently asked questions
What is the fundamental counting principle?+
If one choice can be made in m ways and a second independent choice in n ways, the pair can be made in m × n ways. It extends to any number of stages by multiplying all the counts.
How many outfits from 3 shirts, 4 pants, and 2 shoes?+
3 × 4 × 2 = 24.
How many 4-digit PINs are there?+
10 choices for each of 4 digits: 10^4 = 10,000. If no digit may repeat, 10 × 9 × 8 × 7 = 5,040.
What is the difference between with and without repetition?+
With repetition each position keeps all n options (n^k). Without repetition each used symbol is removed, so the options fall: n × (n-1) × … = n!/(n-k)!.
How does this relate to permutations and combinations?+
The falling product without repetition is exactly the permutation P(n, k). Dividing by k! removes the order and gives the combination C(n, k).
When does the counting principle not apply?+
When the number of options at a stage depends on earlier choices in an irregular way, or when outcomes that differ only in order should count once. Then use casework, permutations with restrictions, or combinations.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026