Combinations and permutations calculator
Math & numbersHow many ways can you pick them?
Exact counts of combinations and permutations, with or without repetition, and the chance of one particular pick.
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Combinations
13,983,816
Ways to choose 6 of 49 items when the order does not matter and nothing repeats.
More about the count
- Formula
- C(49, 6) = 49! ÷ (6! × 43!)
- Chance of one particular pick
- 1 in 13,983,816 (0.00000715%)
- If the order mattered
- 10,068,347,520
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Exact counts of combinations and permutations, with or without repetition, and the chance of one particular pick.
How to use
- Enter how many items there are to pick from and how many you pick.
- Switch on Order matters if a different order is a different outcome, and Repeats allowed if an item can be picked again.
- Read the count as the answer, with the formula, the chance of one particular pick and, for huge counts, every digit.
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How it works
Counting the ways to pick things comes down to two questions: does the order matter, and can an item be picked again? A lottery draw ignores order and never repeats a ball, a race podium cares about order, and a PIN code cares about order and lets digits repeat.
Without repeats, the number of ordered picks is P(n, k) = n! ÷ (n − k)!, where n! = 1 × 2 × … × n. Ignoring the order divides that by the k! ways to arrange the same picks: C(n, k) = n! ÷ (k! × (n − k)!). Choosing 6 numbers out of 49 gives 49! ÷ (6! × 43!) = 13,983,816 rows. With repeats, ordered picks are nᵏ (a four-digit PIN has 10⁴ = 10,000) and unordered picks are C(n + k − 1, k). The formulas follow Graham, Knuth and Patashnik's Concrete Mathematics.
Counts grow very fast, so every digit is computed exactly and big results are also shown as a power of ten. The chance of one particular pick is 1 divided by the count, which holds only when every outcome is equally likely, as in a fair draw.
Worked example
How many pairs can you choose from 5 distinct items without repeats?
- There are 20 ordered picks, but each pair appears in 2 orders.
Ignoring order gives 20 ÷ 2 = 10 pairs.
Frequently asked questions
What is the difference between a combination and a permutation?
In a combination only which items are picked counts; in a permutation the order counts too. Every combination of k items can be arranged in k! orders, so there are k! times as many permutations.
How many possible PIN codes are there?
Turn on both switches and pick 4 of 10 digits: 10⁴ = 10,000 codes. Six digits give 1,000,000.
How many hands of 5 cards can a deck of 52 give?
The order of a hand doesn't matter and a card can't repeat, so it is C(52, 5) = 2,598,960 hands.
Why is the count 0?
Without repeats you cannot pick more items than there are: choosing 7 out of 5 has no way to do it.
Why is choosing no items counted as one way?
There is one empty selection. That is why choosing zero items gives one outcome, even when there are no items available.
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