Permutation Calculator

Calculate permutations and combinations. Find P(n,r) and C(n,r) with step-by-step explanations and examples.

Results

Permutations

720

P(10, 3)

Combinations

120

C(10, 3)

n!3,628,800

Input Values

Results

Permutations (Order Matters)

Ways to arrange 3 items from 10, where order matters

720

P(10, 3)

Combinations (Order Doesn't Matter)

Ways to choose 3 items from 10, where order doesn't matter

120

C(10, 3)

Step-by-Step Calculation

Permutation Formula

P(n, r) = n! / (n - r)!

P(10, 3) = 10! / (10 - 3)!

P(10, 3) = 10! / 7!

P(10, 3) = 3,628,800 / 5,040

P(10, 3) = 720

Combination Formula

C(n, r) = n! / (r! × (n - r)!)

C(10, 3) = 10! / (3! × (10 - 3)!)

C(10, 3) = 10! / (3! × 7!)

C(10, 3) = 3,628,800 / (6 × 5,040)

C(10, 3) = 120

Factorial Reference Table

0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5,040
8! = 40,320
9! = 362,880
10! = 3,628,800
11! = 39,916,800
12! = 479,001,600
13! = 6,227,020,800
14! = 87,178,291,200
15! = 1,307,674,368,000
16! = 20,922,789,888,000
17! = 355,687,428,096,000
18! = 6,402,373,705,728,000
19! = 121,645,100,408,832,000
20! = 2,432,902,008,176,640,000

Permutation vs Combination

Permutation (P)

  • • Order matters
  • • ABC ≠ BAC ≠ CAB
  • • Formula: n!/(n-r)!
  • • Example: Race finishing order
  • • Example: Password arrangements

Combination (C)

  • • Order doesn't matter
  • • ABC = BAC = CAB
  • • Formula: n!/(r!(n-r)!)
  • • Example: Committee selection
  • • Example: Lottery numbers

Key insight: P(n,r) = C(n,r) × r! because each combination can be arranged r! ways to form permutations.

720 = 120 × 6

Real-World Examples

Permutation Examples

  • • Race results: P(8,3) = 336 ways for gold, silver, bronze
  • • PIN codes: P(10,4) = 5,040 four-digit PINs (no repeats)
  • • Book arrangement: 5! = 120 ways to arrange 5 books

Combination Examples

  • • Lottery: C(49,6) = 13,983,816 combinations
  • • Committee: C(10,4) = 210 ways to choose 4 from 10
  • • Poker hand: C(52,5) = 2,598,960 five-card hands

Results

Permutations

720

P(10, 3)

Combinations

120

C(10, 3)

n!3,628,800

?How Do You Calculate Permutations?

Permutations count ordered arrangements where order matters: nPr = n!/(n-r)!. Combinations count selections where order does not matter: nCr = n!/((n-r)! x r!). For example, choosing 3 from 5: permutations = 60 (order matters), combinations = 10 (order does not matter). Use permutations for arrangements, combinations for selections.

What is a Permutation?

Permutations and combinations are counting techniques in combinatorics. Permutations count the number of ways to arrange items where order matters (like passwords or race positions). Combinations count the number of ways to select items where order does not matter (like lottery numbers or committee members). Both use factorial notation.

Key Facts About Permutations

  • Permutation (nPr): order matters. Formula: n!/(n-r)!
  • Combination (nCr): order does not matter. Formula: n!/((n-r)! x r!)
  • nPr is always >= nCr (permutations count each arrangement separately)
  • nCr = nPr / r! (combinations remove the r! arrangements of each selection)
  • 0! = 1 by definition
  • nC0 = nCn = 1 (one way to choose nothing or everything)
  • nC1 = n (n ways to choose 1 item)
  • Pascal's triangle shows nCr values: nCr = (n-1)C(r-1) + (n-1)Cr

Quick Answer

Permutations count ordered arrangements where order matters: nPr = n!/(n-r)!. Combinations count selections where order does not matter: nCr = n!/((n-r)! x r!). For example, choosing 3 from 5: permutations = 60 (order matters), combinations = 10 (order does not matter). Use permutations for arrangements, combinations for selections.

Frequently Asked Questions

A permutation is an arrangement of objects where ORDER MATTERS. P(n,r) counts ways to arrange r items from n total items. Example: Arranging 3 books on a shelf from 5 books = P(5,3) = 60 different arrangements.
A combination is a selection of objects where ORDER DOESN'T MATTER. C(n,r) counts ways to choose r items from n items. Example: Choosing 3 people from 5 for a committee = C(5,3) = 10 different groups.
Permutation: ORDER MATTERS (ABC ≠ BAC). Use for rankings, arrangements, sequences. Combination: ORDER DOESN'T MATTER (ABC = BAC). Use for selections, committees, groups. P(n,r) is always ≥ C(n,r).
Factorial (n!) is the product of all positive integers up to n. 5! = 5×4×3×2×1 = 120. It counts all possible arrangements of n distinct objects. By definition, 0! = 1.
Use permutations when: arranging items in order, assigning positions, creating passwords. Use combinations when: forming committees, choosing lottery numbers, selecting items from a menu.

Last updated: 2025-01-15