Combination Calculator
This combination calculator counts the ways to choose k items from n – depending on whether order matters and whether items can repeat – and shows the formula and working.
Example: n – number of items to choose from 49 · k – number of items chosen 6 · Does order matter: No – only the selection counts (combination) · Can items repeat: No – each item at most once → Number of possibilities: 13,983,816. Source: Wolfram MathWorld – Binomial coefficient. Updated: .
How it is calculated
The number of ways to pick k items from n depends on two questions: does order matter (permutation) or not (combination), and can an item be picked more than once (with repetition)? That gives four formulas.
The four formulas
- Permutation without repetition (nPr): n! ÷ (n − k)! – podium places, seating plans.
- Permutation with repetition: nᵏ – combination locks, PINs, passwords.
- Combination without repetition (nCr, “n choose k”): n! ÷ (k! (n − k)!) – lottery draws, picking a team.
- Combination with repetition: (n + k − 1)! ÷ (k! (n − 1)!) – ice-cream scoops, buying several items from a menu.
Arranging all n items (k = n) gives n! permutations: 5 books on a shelf can be ordered in 5! = 120 ways.
Example: lottery odds
In a 6/49 draw the order does not matter and each ball appears once: 49 choose 6 = (49 · 48 · 47 · 46 · 45 · 44) ÷ (6 · 5 · 4 · 3 · 2 · 1) = 13,983,816. The UK Lotto draws 6 from 59: 45,057,474 combinations, so a single ticket has a 1 in 45 million chance of the jackpot.
Very large numbers
Factorials grow extremely fast – 20! already has 19 digits and 100! has 158. The calculator works with whole numbers of any length and shows huge results in powers of ten as well, just like scientific notation. The nCr and n! keys are also on the scientific calculator, and random draws come from the random number generator.
Frequently asked questions
What does nCr mean?
nCr (“n choose k”) is the number of ways to choose r items from n when order does not matter and nothing repeats: n! ÷ (r! (n − r)!). For example 5C2 = 10.
What is the difference between a permutation and a combination?
In a permutation the order matters (AB ≠ BA); in a combination it does not (AB = BA). So there are always at least as many permutations as combinations.
How many combinations are there in a 6/49 lottery?
13,983,816 – that is 49 choose 6. The chance of matching all six numbers with one ticket is 1 in 13,983,816.
How many 4-digit PINs are there?
10⁴ = 10,000 (permutation with repetition, digits 0–9). Without repeated digits it would be 10 · 9 · 8 · 7 = 5,040.
How do you calculate combinations with repetition?
Use (n + k − 1) choose k. Three scoops from five flavours: 7 choose 3 = 35 possible cups.
What are 0! and n choose 0?
Both equal 1: there is exactly one way to choose nothing or arrange nothing.
Sources and legal basis
- Wolfram MathWorld – Binomial coefficient
- Wolfram MathWorld – Combination
- Wolfram MathWorld – Permutation
- Wolfram MathWorld – Multichoose (combinations with repetition)
As of: · Editorial policy
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