Combinations with Repetition Calculator

What the combinations with repetition calculator does

A combination is a way of choosing some items from a larger set where the order of your choices does not matter. This calculator counts how many combinations are possible, and it handles both cases: when each item can be picked only once, and when the same item can be picked again and again.

Counting choices is the backbone of probability, and doing it by hand means wrestling with factorials. Below is how it works and which case fits your problem.

How to use it

1. Enter the number of objects to choose from, and how many you want to choose.
2. Tick "with repetition" if the same item can be chosen more than once.
3. Press Calculate for the number of combinations, or Reset to clear it.

Combinations mean order does not matter

The defining feature of a combination is that rearranging your picks does not make a new one. Choosing apples, then oranges, then pears gives the same fruit basket as pears, apples, oranges. Only which items you ended up with counts, not the sequence you picked them in.

This is what separates a combination from a permutation. If the order does matter, a lottery draw where positions count, a race finish, a password, then you want permutations instead. When order is irrelevant, as with a committee, a hand of cards, or a basket of goods, combinations are the right count.

How the two cases are worked out

Without repetition, the count is the classic "n choose r", worked out from factorials of the number of objects and the number chosen:

Combinations = n! ÷ ( r! × (n minus r)! )

With repetition, where the same item can be picked again, the formula shifts to allow for those repeats, becoming the number of ways to choose r items from n types when duplicates are allowed. Both are built from factorials, which is exactly the arithmetic that grows unwieldy by hand and where the calculator earns its keep.

With repetition and without

Without repetition is the everyday case: once an item is chosen, it is gone. Dealing a hand of cards, picking a committee from a group of people, choosing lottery numbers that cannot repeat. Each item appears at most once in your selection.

With repetition applies when the same type can be chosen over and over. Scooping three scoops of ice cream from a set of flavours, where all three could be chocolate, is the classic picture. Because repeats are allowed, this case can let you choose more items than there are types, which is impossible without repetition.

A worked example

Suppose an ice cream shop has 5 flavours and you are choosing 3 scoops. If the scoops must all be different, that is combinations without repetition: 5! ÷ (3! × 2!) = 10 possible selections.

But if scoops can repeat, so all three could be the same flavour, that is combinations with repetition, and the count rises to 35. The jump from 10 to 35 is the effect of allowing repeats, since many more baskets become possible once you can pick a flavour more than once.

Entering your values

Enter whole numbers for the objects and the number to choose. When repetition is off, you cannot choose more items than there are objects. When repetition is on, you can, since items may repeat. Because the count is built from factorials, which grow enormous quickly, very large inputs push past what can be shown exactly.

Questions people ask

What is a combination?

A selection of items where order does not matter. Choosing the same items in a different sequence counts as the same combination.

How is a combination different from a permutation?

In a combination the order is irrelevant; in a permutation it matters. A committee is a combination, a ranked finish is a permutation.

When should I use with repetition?

When the same item can be chosen more than once, like scoops of ice cream that can repeat. Without repetition, each item can be picked at most once.

Can I choose more items than there are objects?

Only with repetition, where items may repeat. Without repetition, choosing more than exist is impossible, so the calculator does not allow it.

References

A quick note on where the methods here come from. Combinations and the counting rules of probability are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering combinations and counting.

1. NIST/SEMATECH e-Handbook of Statistical Methods (counting and combinations). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (counting principles and combinations). https://openstax.org/details/books/introductory-statistics-2e