Permutation and Combination Calculator (nPr, nCr & Factorial)

This free permutation and combination calculator finds nPr and nCr together, with and without repetition, plus factorials, circular permutations and arrangements of words with repeated letters (such as MISSISSIPPI). You get the exact integer (not a rounded decimal), the full digits of huge values with scientific notation, the cancelled step-by-step formula using your numbers, an optional list of all arrangements or selections, and the exact TI-84 keystrokes from the MATH > PRB menu. Everything runs in your browser and updates as you type.

Contents

How to Use the Permutation and Combination Calculator

1. Enter n and r. n is how many items you are choosing from; r is how many you pick or arrange. The default example is n = 10, r = 3.
2. Read all four counts at once. nPr (order matters, no repeats), nCr (order does not matter, no repeats), n^r (order matters, repeats allowed) and C(n + r − 1, r) (order does not matter, repeats allowed).
3. Open the steps and TI-84 keys. The cancelled product uses your numbers. The PRB keystrokes show what a TI-84 should display, or OVERFLOW if the answer is larger than about 1 × 10¹⁰⁰.
4. Optional modes. List items types a set such as A, B, C, D, E and lists the arrangements or selections (capped at 5,000). Repeated letters counts distinct arrangements of a word. Circular counts seatings around a table as (n − 1)!.

Copy share link stores your inputs in the URL so you can send the same problem to a classmate. For the nCr term that appears in a probability mass function, see the binomial distribution calculator.

Permutation vs Combination: Does Order Matter?

One question decides which formula to use: does the order matter?

If you are counting…	Order?	Use	Tiny example (n = 10, r = 3)
Rankings, codes, podium places, passwords, assigned roles (chair, treasurer, secretary)	Yes	Permutation nPr	10P3 = 720
Teams, committees, lottery tickets, hands of cards, toppings on a pizza	No	Combination nCr	10C3 = 120

Every combination of r items can be arranged in r! different orders, which is why nPr = nCr × r!. In the default example, 120 × 3! = 120 × 6 = 720.

Permutation Formula (nPr)

A permutation counts ordered arrangements of r items taken from n distinct items, with no item used twice:

nPr = P(n, r) = n! / (n − r)! = n × (n − 1) × … × (n − r + 1)

You never need the full n! if r is smaller than n: cancel (n − r)! and multiply only the first r factors. Arranging all n items is nPn = n!.

Combination Formula (nCr)

A combination counts unordered selections — “n choose r”, also written C(n, r) or as a binomial coefficient:

nCr = C(n, r) = n! / (r! (n − r)!) = nPr / r!

This nCr is the same coefficient that appears in the binomial theorem and in the binomial probability P(X = x) = nCr · p^x · (1 − p)^{n − x}.

Worked Example: 10P3 and 10C3

The calculator opens with n = 10 and r = 3, a standard textbook pair: gold, silver and bronze among 10 runners, versus a 3-person committee from 10 people.

1. Permutations. 10P3 = 10! / 7! = 10 × 9 × 8 = 720. There are 10 choices for gold, then 9 for silver, then 8 for bronze.
2. Combinations. 10C3 = 10P3 / 3! = 720 / 6 = 120. Written as a cancelled fraction: (10 × 9 × 8) / (3 × 2 × 1) = 120. The committee {A, B, C} is the same as {C, A, B}, so the 6 orderings of each group are counted once.
3. With repetition. A 3-character code from 10 symbols with repeats allowed is 10³ = 1,000. Choosing 3 scoops from 10 flavours when doubles are allowed is C(10 + 3 − 1, 3) = C(12, 3) = 220.
4. Factorials. 10! = 3,628,800, 3! = 6 and 7! = 5,040. Check: 120 × 6 = 720.
5. Probability. If one of the 120 committees is equally likely, the chance of one particular committee is 1 / 120 ≈ 0.008333.

With and Without Repetition

The usual nPr and nCr formulas assume you cannot pick the same item twice. When replacement is allowed, the formulas change. The four cases together are:

	No repetition	With repetition
Order matters	nPr = n! / (n − r)!	nr
Order does not matter	nCr = n! / (r! (n − r)!)	C(n + r − 1, r)

A 4-digit PIN (digits 0–9, repeats allowed, order matters) is 10⁴ = 10,000. A lottery ticket that draws 6 different numbers from 49 is 49C6 = 13,983,816. Ice-cream scoops where you may repeat a flavour are a combination with repetition (“stars and bars”). r may be larger than n only in the repetition cases: you cannot choose 6 different people from a group of 5, but you can make a 6-digit code from 5 symbols.

Factorial (n!) and Why 0! = 1

n! (n factorial) is the product n × (n − 1) × … × 1, and it equals nPn, the number of ways to line up n distinct items. The calculator reports n!, r! and (n − r)! next to the main counts.

0! = 1, by definition. There is one way to arrange an empty list (do nothing), and the formulas only stay consistent if 0! is 1: nCn = n! / (n! 0!) = 1, and nP0 = nC0 = 1.

Circular Permutations

When people sit around a table, rotating everyone one seat is the same arrangement, so you divide the linear count n! by n:

Circular permutations = (n − 1)!

The default circular example is n = 8 people: 7! = 5,040 distinct seatings. If a reflection also counts as the same (a necklace or bracelet that can be flipped, or a see-through table with no clockwise/counterclockwise difference), divide by 2 to get (n − 1)! / 2. For n = 8 that is 2,520. Tick the reflections box only when a flip should not count as new.

Permutations with Repeated Letters (Multinomial)

For a word with repeated letters the distinct arrangements are given by the multinomial coefficient:

n! / (n₁! n₂! … n_k!)

where n is the total number of letters and the n_i are the frequencies of each distinct letter. The classic example MISSISSIPPI has 11 letters: M × 1, I × 4, S × 4, P × 2.

11! / (4! × 4! × 2! × 1!) = 39,916,800 / (24 × 24 × 2) = 34,650.

If every letter were distinct there would be 11! arrangements; swapping two identical S’s would not make a new word, so we divide by 4! for the S’s, 4! for the I’s and 2! for the P’s. Type any word in the Repeated letters tab to see the frequencies and the cancelled working.

How to Do nPr and nCr on a TI-84

1. nPr: type n, press MATH, move to the PRB menu (left arrow), choose 2:nPr, type r, press ENTER. For the default example: 10 nPr 3 gives 720.
2. nCr: the same path, but 3:nCr. 10 nCr 3 gives 120.
3. Factorial: type n, MATH PRB 4:!, ENTER. 10! gives 3628800.
4. OVERFLOW: a TI-84 can only display results up to about 9.999999999 × 10⁹⁹. Anything at or above 1 × 10¹⁰⁰ shows OVERFLOW. 69! still fits (about 1.71 × 10⁹⁸); 70! already overflows. The calculator above still shows the exact integer and warns you when a handheld would overflow.
5. With repetition: the TI-84 has no special command. Use the ^ key for n^r, or compute C(n + r − 1, r) with nCr. For circular permutations compute (n − 1)!. For repeated letters, divide n! by each frequency factorial.

The results panel prints these keystrokes with your own n and r. Practise them on our TI-84 Plus CE calculator online. The PRB menu is also on the TI-30XS MultiView. Browse every tool on the math calculators hub.

Common Mistakes

• Using nCr when roles are different. A chair, a treasurer and a secretary are different jobs, so order (or role) matters: use nPr, not nCr.
• Forgetting that repeats are allowed. PINs, passwords and licence plates usually allow repeated symbols. Using nPr undercounts, often by a lot.
• Writing nPr = n! / r!. The correct denominator is (n − r)!, not r!. nCr is the one that also divides by r!.
• Thinking 0! is 0. 0! = 1, and nC0 = nP0 = 1.
• r > n without repetition. You cannot choose 6 different people from 5. You can only have r > n when repetition is allowed.
• Taking OVERFLOW as “the answer is impossible.” The handheld just cannot display an integer that large. The exact value still exists, and this calculator will show it (up to 8,000 digits).
• Using n! for a round table. Rotations are the same seating, so use (n − 1)!.
• Arranging MISSISSIPPI as 11!. Identical letters must be divided out or you overcount.

Why This Permutation and Combination Calculator Is Different

• nPr and nCr together, plus both repetition modes (n^r and C(n + r − 1, r)), factorials, circular permutations and the multinomial for repeated letters.
• Exact integers with BigInt, not floating-point rounding. Values such as 1000! (2,568 digits) are computed in full, with scientific notation and a copyable digit listing. Nothing is rounded off until a result would exceed 8,000 digits.
• Step-by-step cancellation using your n and r, so you can see 10 × 9 × 8 / (3 × 2 × 1) rather than a bare 120.
• TI-84 keystrokes with your numbers for 2:nPr, 3:nCr and 4:!, including the OVERFLOW limit (results above about 1 × 10¹⁰⁰). No top-ranking permutation calculator we checked does this.
• Optional listings of arrangements and selections from a list you type, capped at 5,000 results.
• Share links, dark mode and a phone-friendly layout, with clear errors for r > n, negatives, non-integers and oversized inputs.
• Checked against Python’s math.perm, math.comb, math.factorial and itertools on hundreds of random and edge cases, including 0, 1, n = r, r = 0 and n = 1,000.

Permutation and Combination Calculator FAQ

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