Combination without Repetition Calculator

Use the formula C(n,r) = n! / [r! × (n−r)!], where n is the total number of objects and r is the sample size. Compute the factorial of n, divide it by the product of r! and (n−r)!, and the result is the number of ways to choose r items from n without repetition and without regard to order.

A combination counts the number of ways to choose r elements from n where order does NOT matter. A permutation counts arrangements where order DOES matter. For example, choosing {1, 2, 3} and {3, 2, 1} is the same combination but two different permutations.

'Without repetition' (also called without replacement) means each element can be chosen at most once. For instance, when picking lottery numbers you cannot pick the same number twice — that is a combination without repetition.

Using C(10,4) = 10! / (4! × 6!) = 210. So there are 210 unique ways to choose 4 numbers from a set of 10 when order does not matter and repetition is not allowed.

It depends on the total pool size n. For example, C(5,2) = 10, C(10,5) = 252, and C(20,5) = 15,504. Enter your specific values of n and r into the calculator to get the exact count.

Again, this depends on the sample size r. For C(16,4) = 1,820 and C(16,8) = 12,870. Use the calculator above with n = 16 and your desired r to find the exact number.

No. If r is greater than n, it is mathematically impossible to choose r distinct elements from a set of only n objects. In that case, the number of combinations is 0 and the formula is undefined.

When r = 0, there is exactly 1 combination — the empty selection. When r = n, there is also exactly 1 combination — choosing every element in the set. Both edge cases evaluate to C = 1 using the formula.