The formula this page uses
P(n, r) = n! / (n − r)! C(n, r) = n! / ( r! · (n − r)! ) |A ∪ B| = |A| + |B| − |A ∩ B|
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Discrete Math Calculator
Solve discrete math problems with clear steps, notation, and a final check.
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Evaluate the governing formula locally in your browser. Define each known quantity once, then change values to test another case instantly.
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Discrete math
Discrete Math Calculator explained
The short version
- Discrete math counts things you can list: arrangements, subsets, routes, and truth values.
- The first question is always whether order matters. Order matters means permutation, order does not means combination.
- The second question is whether repeats are allowed, and that changes the formula again.
What each part means
SymbolWhat it means
n — Size of the poolHow many different items are available to pick from.
r — How many you pickMust be no larger than n unless repeats are allowed.
n! — Factorialn × (n−1) × … × 1, and 0! is defined to be 1.
|A ∩ B| — OverlapHow many items are in both sets, which is what inclusion-exclusion subtracts once.
Show your work: a full example
- The setup10 people, choose 3
- If the three jobs are different, order mattersP(10, 3) = 10 × 9 × 8
- Multiply720 ordered ways
- If they are just a committee, order does not mattereach committee of 3 was counted 3! = 6 times
- So divideC(10, 3) = 720 ÷ 6
- The answer120 committees
- Read itthe same 3 people give 6 different podium orders but only 1 committee, which is exactly the factor of 6
A second, different case
- A different case: counting overlapping groups100 students, 60 take maths, 45 take physics, 25 take both
- Adding naively double-counts the overlap60 + 45 = 105, which is already more than the 100 students
- Inclusion-exclusion fixes it|M ∪ P| = 60 + 45 − 25
- At least one subject80 students
- Neither subject100 − 80 = 20
- Only maths, and only physics60 − 25 = 35 and 45 − 25 = 20
- Check the four groups add back35 + 25 + 20 + 20 = 100 ✓
Copy-ready example
10 people, choose 3
The setup
Questions about the Discrete Math Calculator
How do I decide between a permutation and a combination?
Ask whether swapping two chosen items gives a genuinely different outcome. Gold and silver medals: yes, so permutation. Two seats on the same committee: no, so combination. The combination is always the smaller number.
Why is 0! defined as 1?
Because there is exactly one way to arrange nothing, and because it keeps the formulas working. C(10, 10) should be 1, and 10!/(10!·0!) only gives 1 if 0! = 1.
What is the pigeonhole principle?
If you put more items than boxes, some box holds at least two. With 13 people, two must share a birth month, because there are only 12 months. It proves things exist without ever finding them.
Where is discrete math actually used?
Anywhere the objects come in whole units: hashing and password strength use n^r, network routing uses graphs, and program correctness uses logic and induction. Continuous calculus has almost nothing to say about any of it.