Last updated
Basic Permutation Calculation
P(n, r) — arrangements where order matters:
P(5, 3) — Arrange 3 books from 5 on a shelf
Formula: P(n,r) = n! / (n-r)!
P(5,3) = 5! / (5-3)!
= 5! / 2!
= (5 × 4 × 3 × 2 × 1) / (2 × 1)
= 120 / 2
= 60
Answer: 60 different arrangements
Order matters here — Book A, B, C on the shelf is different from Book C, B, A.
Basic Combination Calculation
C(n, r) — selections where order doesn't matter:
C(5, 3) — Choose 3 books from 5 for a reading list
Formula: C(n,r) = n! / (r! × (n-r)!)
C(5,3) = 5! / (3! × 2!)
= 120 / (6 × 2)
= 120 / 12
= 10
Answer: 10 different selections
Order doesn't matter — {A, B, C} is the same selection as {C, B, A}. That's why C(5,3) = 10 is much less than P(5,3) = 60.
Permutation vs Combination — Side by Side
n=5, r=3:
P(5,3) = 60 (order matters — arrangements)
C(5,3) = 10 (order doesn't matter — selections)
Relationship: C(n,r) = P(n,r) / r!
C(5,3) = 60 / 3! = 60 / 6 = 10
The r! factor accounts for the r! ways to arrange the same r items.
Each combination corresponds to r! permutations.
Lottery Probability
Calculating the odds of winning a lottery:
Pick 6 numbers from 1-49 (order doesn't matter):
C(49, 6) = 49! / (6! × 43!)
= (49 × 48 × 47 × 46 × 45 × 44) / (6 × 5 × 4 × 3 × 2 × 1)
= 10,068,347,520 / 720
= 13,983,816
Probability of winning: 1 in 13,983,816 ≈ 0.0000072%
Pick 5 from 69 + 1 from 26 (Powerball):
C(69,5) × C(26,1) = 11,238,513 × 26 = 292,201,338
Probability: 1 in 292,201,338
Password Permutations
How many possible passwords exist for a given configuration?
8-character password, lowercase only (26 chars):
26^8 = 208,827,064,576 ≈ 208 billion
8-character password, alphanumeric (62 chars):
62^8 = 218,340,105,584,896 ≈ 218 trillion
12-character password, all printable ASCII (94 chars):
94^12 = 475,920,314,814,253,376,475,136 ≈ 4.76 × 10^23
Note: These are permutations with repetition (characters can repeat).
Formula: n^r (not P(n,r) which doesn't allow repetition)
Card Game Combinations
Counting poker hands:
Total 5-card hands from 52-card deck:
C(52, 5) = 2,598,960
Royal Flush (10, J, Q, K, A of same suit):
4 possible (one per suit)
Probability: 4 / 2,598,960 = 0.000154%
Four of a Kind:
C(13,1) × C(4,4) × C(48,1) = 13 × 1 × 48 = 624
Probability: 624 / 2,598,960 = 0.024%
Full House (3 of a kind + pair):
C(13,1) × C(4,3) × C(12,1) × C(4,2)
= 13 × 4 × 12 × 6 = 3,744
Probability: 3,744 / 2,598,960 = 0.144%
Team Selection
Choosing a committee from a group:
Select a 5-person committee from 20 candidates:
C(20, 5) = 20! / (5! × 15!)
= (20 × 19 × 18 × 17 × 16) / (5 × 4 × 3 × 2 × 1)
= 1,860,480 / 120
= 15,504 possible committees
Select a president, VP, and secretary from 20 people (order matters):
P(20, 3) = 20! / 17!
= 20 × 19 × 18
= 6,840 possible arrangements
Race Podium Permutations
How many ways can 3 medals be awarded to 10 runners?
P(10, 3) = 10! / 7!
= 10 × 9 × 8
= 720
Gold, Silver, Bronze can be awarded in 720 different ways.
If we only care about who wins (not which medal):
C(10, 3) = 720 / 3! = 720 / 6 = 120
120 different sets of 3 winners
Large Number Calculations
The calculator handles exact arithmetic for large inputs:
C(100, 50) = exact result:
100891344545564193334812497256
P(50, 20) = exact result:
31469973260387937525653122354950764088012280797258232192163168247821107200000000000000
Standard calculators lose precision at these sizes.
This calculator uses arbitrary precision arithmetic for exact results.
Special Cases
P(n, 0) = 1 (one way to arrange zero items: do nothing)
P(n, 1) = n (n ways to choose one item)
P(n, n) = n! (n! ways to arrange all n items)
C(n, 0) = 1 (one way to choose zero items)
C(n, 1) = n (n ways to choose one item)
C(n, n) = 1 (one way to choose all items)
C(n, r) = C(n, n-r) (choosing r is same as excluding n-r)
Example: C(10, 3) = C(10, 7) = 120
Choosing 3 from 10 is equivalent to choosing which 7 to leave out.