Combination Calculator

What is a combination?

A combination is a way of picking items from a larger set when the order of the picks does not matter. A committee made of Ana, Ben and Chloe is the same committee whether you list the names as Ana, Ben, Chloe or Chloe, Ana, Ben. Only the group counts, not the sequence in which it was assembled.

The number of combinations of k items taken from n is written C(n, k), nCr, or "n choose k". It is also called a binomial coefficient, because the same numbers appear as the coefficients when a binomial such as (a + b)ⁿ is expanded. When the order of selection does matter you are counting permutations instead; see the permutation calculator.

How to use the calculator

1. Type the number of objects available in Choose from (n).
2. Type how many of them you will pick in How many? (k).
3. Tick With repetition if each pick is made from the full set again, so the same type can appear several times.
4. Press Calculate. The answer appears in the Combinations row.

Both inputs must be whole numbers of at least 1. The tool works in ordinary double-precision arithmetic, so a count with more than about 15 significant digits is the nearest number the computer can represent rather than an exact integer, and a factorial above 170! overflows, which the calculator reports as inf or nan. That limit is reached when n is larger than 170, or when n + k − 1 is larger than 170 with repetition.

Without repetition you cannot choose more items than exist, so the true count for k greater than n is zero. The formula is still evaluated with the convention that the factorial of a negative number is 1, which is why the display can show a small fraction such as 0.05 instead of 0 in that case. Read any such result as "no valid selection".

Combination formulas

Without repetition, each selection of k items could have been made in k! different orders. Counting ordered picks and then dividing by k! removes that duplication:

C(n, k) = n! ÷ (k! × (n − k)!)n objects, k chosen, no object used twice.

When repetition is allowed, think of distributing k identical picks among n types. The standard "stars and bars" argument shows that this equals choosing k positions out of n + k − 1:

C(n + k − 1, k) = (n + k − 1)! ÷ (k! × (n − 1)!)n types available, k picks, any type may repeat.

Worked examples

Choose 3 from 5. C(5, 3) = 5! ÷ (3! × 2!) = 120 ÷ (6 × 2) = 10. The ten groups of three from the letters A to E are ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE and CDE.

A 6-from-49 lottery. Cancelling most of the factorial gives 49 × 48 × 47 × 46 × 45 × 44 ÷ 720 = 13,983,816 possible tickets, so one ticket has a chance of about 1 in 14 million.

Five-card hands. A deck of 52 cards offers C(52, 5) = 2,598,960 different hands, regardless of the order in which the cards are dealt.

Three scoops from five flavours, flavours may repeat. With repetition the count is C(5 + 3 − 1, 3) = C(7, 3) = 35. Without repetition it would be only 10, because every scoop would have to be a different flavour.

N choose K table

Each number below is the sum of the two numbers above it, which is Pascal's rule: C(n, k) = C(n − 1, k − 1) + C(n − 1, k). Read row n and column k to find n choose k.

Useful properties

• Symmetry: C(n, k) = C(n, n − k). Choosing the 3 people who go is the same as choosing the 2 who stay behind out of 5.
• Edges: C(n, 0) = C(n, n) = 1. There is exactly one way to pick nothing and one way to pick everything.
• Total subsets: adding every C(n, k) from k = 0 to n gives 2ⁿ. For n = 5 the row is 1 + 5 + 10 + 10 + 5 + 1 = 32 = 2⁵.
• Permutations: the number of ordered selections is P(n, k) = C(n, k) × k!. For 3 from 5 that is 10 × 6 = 60, the same as 5 × 4 × 3.

Common mistakes

• Using nCr when the order of the picks matters, for example a race podium or a PIN code. Those are permutations.
• Forgetting repetition. Choosing a basket of fruit where you can take several apples is a with-repetition problem; dealing cards from one deck is not.
• Swapping n and k. The total comes first, the number you pick second.
• Expecting exact integers for very large inputs. Beyond about 15 digits the figure is rounded; for exact work use big-integer arithmetic.