Free Probability Calculator — Combinations, Permutations & Factorials

What Is a Probability Calculator?

Counting problems appear everywhere — how many ways can you choose a committee, arrange people in a line, deal a poker hand, or create a password? These questions all come down to three fundamental operations: combinations (choosing without regard to order), permutations (arranging where order matters), and factorials (counting all possible orderings). The formulas involve large factorials that grow explosively, making manual computation impractical.

Our free probability calculator computes combinations (nCr), permutations (nPr), and factorials (n!) instantly. It doesn't just give you the answer — it shows the formula being used and the step-by-step expansion so you can follow the mathematics and understand why the result is what it is.

Common Use Cases

• Combinations (nCr) - calculate how many ways to choose r items from n when order doesn't matter
• Permutations (nPr) - calculate how many ways to arrange r items from n when order matters
• Factorials (n!) - compute the total number of ways to arrange n items in sequence
• Step-by-step formula expansion - see exactly how the formula is applied with your specific numbers
• Large number computation - handle factorials and combinations involving large values that overflow hand calculations
• Formula reference - see the exact formula (nCr = n!/(r!(n-r)!)) applied to your inputs as a learning tool

Why Counting Problems Are Challenging

• Combinations vs. permutations confusion - the hardest part is often deciding whether order matters (permutation) or doesn't (combination), and choosing wrong gives a completely different answer
• Factorial growth is explosive - 10! is already 3,628,800 and 20! is over 2 quintillion; hand computation becomes impossible very quickly
• Formula confusion is common - remembering whether nCr uses r!(n-r)! or (n-r)! in the denominator, or whether nPr divides by (n-r)! or r!, trips up even experienced students
• Arithmetic overflow in intermediate steps - even when the final answer is reasonable, the intermediate factorials can be astronomically large, causing errors in step-by-step hand computation

How It Works

Select the type of calculation — combination, permutation, or factorial. Enter the required values (n and r for combinations and permutations, just n for factorials). The calculator displays the formula, substitutes your values, expands the factorials, cancels common factors, and shows the final result.

Seeing the step-by-step expansion is valuable because it shows how factorial cancellation works — for example, in C(10,3), the 10! in the numerator and 7! in the denominator cancel to leave just 10 × 9 × 8, which is much easier to compute.

Example

Input: n = 10, r = 3

Operation: Combination

Output: C(10, 3) = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120

There are 120 ways to choose 3 items from 10 when order doesn't matter.

Key Examples

• Committee selection - C(8, 3) = 56 ways to choose a 3-person committee from 8 candidates, since the order of selection doesn't matter
• Medal awards - P(5, 3) = 60 ways to award gold, silver, and bronze to 5 athletes, since which person gets which medal matters
• Seating arrangements - 5! = 120 ways to seat 5 people in a row, as every different ordering is a distinct arrangement
• Poker hands - C(52, 5) = 2,598,960 possible 5-card poker hands from a standard deck, the foundation of all poker probability calculations
• 4-letter codes - P(26, 4) = 358,800 possible 4-letter codes from the alphabet with no repeated letters, since the order of letters matters

Benefits

• Instant combinations, permutations, and factorials - get results for counting problems that would take many steps by hand
• See the formula expansion step by step - understand how factorial cancellation works and why the answer is what it is
• Handle large numbers accurately - compute results involving large factorials without overflow errors
• 100% browser-based - no downloads, no accounts, instant results on any device
• Perfect for probability homework - verify your work on counting problems, understand the formulas, and build combinatorial intuition

Frequently Asked Questions

What is the difference between combinations and permutations?

Combinations count selections where order doesn't matter — choosing {A, B, C} is the same as choosing {C, A, B}. Permutations count arrangements where order matters — the arrangement ABC is different from CAB. Permutations always give a larger (or equal) result because they count more distinct outcomes.

What is the maximum number the calculator can handle?

The calculator can handle values of n up to several hundred for combinations and permutations. For factorials, it can compute n! for large values of n accurately. Extremely large results are displayed in scientific notation when they exceed standard number formatting.

What exactly is a factorial?

A factorial (written n!) is the product of all positive integers from 1 to n. So 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1. Factorials count the number of ways to arrange n distinct items in a complete sequence.

When should I use combinations vs. permutations?

Ask yourself: does the order of selection matter? If you're choosing a committee (order doesn't matter), use combinations. If you're assigning ranked positions like 1st, 2nd, 3rd (order matters), use permutations. A helpful test: would rearranging the selected items create a "different" outcome? If yes, use permutations; if no, use combinations.

How accurate are the results for very large numbers?

The calculator uses JavaScript's number precision, which handles integers exactly up to 2⁵³ (about 9 quadrillion). For larger intermediate values, the calculator uses factorial cancellation to keep numbers manageable. Results that exceed standard precision are displayed in scientific notation.

Is the probability calculator free to use?

Yes, the calculator is completely free with no usage limits. Compute as many combinations, permutations, and factorials as you need for probability coursework, statistics, or any counting problem.

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