Permutations & Combinations Calculator

About Permutations & Combinations Calculator

Calculate permutations (nPr) and combinations (nCr) with and without repetition. Enter the total items and items chosen, and see the result with full factorial expansion and step-by-step working. Supports BigInt for exact results even with very large numbers.

Permutations vs Combinations - Which Do I Need?

The distinction comes down to two questions: does order matter, and can items repeat?

	Order Matters (Permutation)	Order Does Not Matter (Combination)
No repetition	nPr = n! / (n-r)!	nCr = n! / (r!(n-r)!)
With repetition	n^r	(n+r-1)! / (r!(n-1)!)

Quick rule: If rearranging the same items gives a different outcome (like a PIN code), use permutations. If the selection is the same regardless of order (like a lottery draw), use combinations.

Permutations Without Repetition (nPr)

Used when you are arranging r items from n, each item used at most once, and the order of arrangement matters.

nPr = n! / (n - r)!

Worked example: How many ways can 3 runners finish first, second, and third from a race of 10?

1. n = 10, r = 3
2. 10P3 = 10! / (10-3)! = 10! / 7!
3. = 10 × 9 × 8 = 720

More examples:

Scenario	n	r	nPr
Arranging 5 books on a shelf from 8	8	5	6,720
Assigning president, VP, secretary from 12	12	3	1,320
4-letter codes from 26 letters (no repeats)	26	4	358,800
Seating 6 people in 6 chairs	6	6	720

Combinations Without Repetition (nCr)

Used when you are selecting r items from n, each item used at most once, and the order does not matter.

nCr = n! / (r! × (n - r)!)

Worked example: How many 5-card poker hands can be dealt from a standard 52-card deck?

1. n = 52, r = 5
2. 52C5 = 52! / (5! × 47!)
3. = (52 × 51 × 50 × 49 × 48) / (5 × 4 × 3 × 2 × 1)
4. = 311,875,200 / 120 = 2,598,960

More examples:

Scenario	n	r	nCr
Choosing 3 toppings from 10	10	3	120
Lottery: 6 numbers from 49	49	6	13,983,816
Selecting a committee of 4 from 15	15	4	1,365
Choosing 2 desserts from a menu of 8	8	2	28

Permutations With Repetition

When items can be reused and order matters, the formula is simply:

n^r

Example: A 4-digit PIN using digits 0-9 (each digit can repeat):

• n = 10, r = 4
• 10^4 = 10,000 possible PINs

Scenario	n	r	n^r
Binary strings of length 8	2	8	256
3-letter codes (A-Z, repeats ok)	26	3	17,576
Coin flips, 10 tosses	2	10	1,024
6-character password (lowercase only)	26	6	308,915,776

Combinations With Repetition

When items can be reused and order does not matter (also called "multiset coefficients"):

(n + r - 1)! / (r! × (n - 1)!)

Example: Choosing 3 scoops from 5 ice cream flavours (same flavour can be picked again):

• n = 5, r = 3
• (5 + 3 - 1)! / (3! × 4!) = 7! / (6 × 24) = 5,040 / 144 = 35

Pascal's Triangle and nCr

Every entry in Pascal's triangle is a combination value. Row n, position r gives nCr:

Row (n)	Values (nC0 through nCn)
0	1
1	1, 1
2	1, 2, 1
3	1, 3, 3, 1
4	1, 4, 6, 4, 1
5	1, 5, 10, 10, 5, 1
6	1, 6, 15, 20, 15, 6, 1

Each number is the sum of the two numbers directly above it. This gives us useful identities like nCr = (n-1)C(r-1) + (n-1)Cr.

Real-World Probability Connection

Combinations are the foundation of probability calculations. The probability of an event is often (favourable outcomes) / (total outcomes), and both counts use nCr.

Example: Probability of winning a 6/49 lottery:

• Total outcomes: 49C6 = 13,983,816
• Winning outcomes: 1
• Probability: 1/13,983,816 ≈ 0.0000000715 (about 1 in 14 million)

For more probability calculations, the probability calculator handles events, conditional probability, and Bayes' theorem. For computing individual factorial values, the factorial calculator provides exact BigInt results.

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