Permutation and Combination Calculator

Suppose 8 sprinters line up for a race and you want to know how many ways the medals could be handed out for 1st, 2nd, and 3rd place. Because finishing order matters, that is a permutation: 8P3 = 336. Now suppose instead you just want to pick 3 of those 8 runners for a relay team, where it does not matter who you name first. That is a combination: 8C3 = 56. Same 8 people, same 3 chosen, but six times as many permutations as combinations - because each team of 3 can be ordered in 3! = 6 ways. That single idea, "does order matter?", is the whole game, and this permutation and combination calculator answers both at once so you can see the gap instantly.

The formulas

Both formulas are built from the factorial, written n!, which is the product of every whole number from 1 up to n (and 0! is defined as 1). The permutation and combination formulas are:

Here n is the total number of distinct items and r is how many you select. The only difference between the two is the extra r! in the denominator of the combination formula. That r! removes the duplicate orderings, which is why the two are linked by the tidy relationship nCr = nPr ÷ r!.

How to use this calculator

You only need two numbers:

1. Enter n - the total number of distinct items you are choosing from (for example, 52 cards or 10 books).
2. Enter r - how many items you are selecting or arranging (for example, 5 cards or 3 books). r must be less than or equal to n.
3. Read both results. The blue card shows permutations (order matters); the cyan card shows combinations (order does not matter). They update instantly as you type.
4. Check the steps. The step-by-step panel substitutes your numbers into each formula, and the reference table lists nPr and nCr for every r at your chosen n.

If you enter a value of r larger than n, or a non-whole number, the calculator explains what to fix instead of showing a wrong answer.

Permutation vs. combination: which one do I need?

Ask yourself a single question: if I shuffle the chosen items, is it a different outcome?

• If yes, use permutations (nPr). Race finishes, passwords and PINs, seating in a row, ranking, assigning specific roles (president vs. treasurer) - order changes the result.
• If no, use combinations (nCr). Lottery numbers, a hand of cards, a committee, a pizza's toppings, choosing a subset - shuffling the picks gives the same group.

Worked example 1 - a deck of cards

How many different 5-card poker hands come from a standard 52-card deck? The order you are dealt the cards does not matter, so this is a combination: 52C5 = 52! ÷ (5! × 47!) = 2,598,960. If order did matter (say you were dealt the cards into 5 labeled slots), it would be the permutation 52P5 = 311,875,200 - exactly 5! = 120 times larger.

Worked example 2 - a lottery

A "6 from 49" lottery draws 6 numbers from 49, and order does not matter, so the number of possible tickets is 49C6 = 13,983,816. That is why the odds of matching all six on one ticket are about 1 in 14 million. The permutation 49P6 is far larger (about 10 billion) because it would treat every drawn order as distinct - which a real lottery does not.

Worked example 3 - arranging vs. choosing books

You have 10 books and a shelf that holds 3. How many ways can you arrange 3 on the shelf? Order matters (left-to-right), so 10P3 = 10 × 9 × 8 = 720. How many distinct sets of 3 books could you pick to take on a trip? Order is irrelevant, so 10C3 = 720 ÷ 3! = 120. The 720 arrangements collapse into 120 sets because each set of 3 can be lined up in 3! = 6 ways.

Key terms

• Factorial (n!): the product of all whole numbers from 1 to n. 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention 0! = 1.
• n: the size of the full set you are choosing from.
• r: the number of items you select or arrange, with 0 ≤ r ≤ n.
• Permutation (nPr): an ordered arrangement of r items from n.
• Combination (nCr): an unordered selection of r items from n; also called the "binomial coefficient" and written as (n choose r).
• Without replacement: each item is used at most once - the assumption behind both formulas on this page.

Quick reference table

A few common values, so you can sanity-check the calculator at a glance:

Useful tips and shortcuts

• Symmetry of combinations: nCr = nC(n − r). Choosing 47 cards to leave behind is the same as choosing 5 to keep, so 52C47 = 52C5. Use the smaller r for quicker mental math.
• Edge cases are easy: nC0 = nCn = 1 (one way to choose nothing or everything), nC1 = n, and nP1 = n.
• Don't compute giant factorials directly: nPr is just the product of r descending terms (n × (n−1) × ...), so you rarely need the full n!. This calculator uses that shortcut internally.
• For probability, the count from this tool is usually the denominator: divide favorable outcomes by total outcomes (often an nCr) to get the odds.

Common pitfalls

• Using the wrong tool: the single most common error is treating an ordered problem as unordered (or vice versa). Always check whether rearranging the picks creates a new outcome.
• Assuming repetition: these formulas are "without replacement." If items can repeat (digits in a PIN, scoops on a cone with repeats allowed), you need n^r or the repetition combination formula instead.
• Forgetting identical items: arranging the letters of "LEVEL" is not 5! because letters repeat; you divide by the factorials of the repeats. That is a different (multiset) formula.
• Mixing up n and r: n is the pool, r is the pick. Swapping them changes the answer and can make r > n, which is invalid.

Related concepts

Permutations and combinations are the foundation of counting and probability. The combination nCr is also the binomial coefficient that appears in Pascal's triangle and the binomial theorem (the coefficients when you expand (a + b)^n). Closely related ideas include selections with repetition (n^r for ordered, (n + r − 1)C r for unordered) and circular permutations, where arranging r distinct items in a ring gives (r − 1)! because rotations are considered the same. Once you are comfortable with nPr and nCr, those variations follow naturally.

How it compares to related calculators

This page answers "how many ways can I arrange or choose r items from n?" For a different question, a sister tool fits better:

• To work with percentages and odds, use the Percentage Calculator.
• To add, reduce, and compare fractions (like simplifying a probability), use the Fraction Calculator.
• For factorials, powers, and general expressions, use the Scientific Calculator.
• To raise numbers to a power (the n^r in repetition problems), use the Exponent Calculator.
• For means and basic statistics, use the Average Calculator.