Combination Calculator
Combination Calculator C(n, r)
About Combination Calculator
What Is a Combination?
A combination is a selection of items where order does not matter. Choosing players A, B, and C for a team is the same combination regardless of the order you picked them. Combinations answer questions like: how many different 5-card hands can be dealt from a 52-card deck? The answer is C(52,5) = 2,598,960. Lottery odds, team selection, committee formation, and statistical sampling all use combinations. The combination formula is also called the binomial coefficient because it appears in the binomial theorem for expanding (a+b)^n.
The Combination Formula
The formula is C(n,r) = n! / (r! x (n-r)!). This is equivalent to P(n,r) / r! because combinations are permutations divided by the number of ways to arrange the selected items (removing order). For example, C(10,3) = 10! / (3! x 7!) = 120. With repetition allowed: C(n+r-1, r) = (n+r-1)! / (r! x (n-1)!).
How to Use This Calculator
Enter n (total objects) and r (objects to choose). Select whether repetition is allowed. Click Calculate to see the number of combinations along with the step-by-step calculation showing the factorial expansion.
Frequently Asked Questions
How do I know whether to use combinations or permutations?
Ask: does the order of selection matter? If choosing members for a committee, order does not matter (combination). If assigning specific roles (president, VP, secretary), order matters (permutation).
What are lottery odds?
Lottery odds use combinations. Picking 6 numbers from 49: C(49,6) = 13,983,816. You have a 1 in 14 million chance of matching all 6 numbers.
What is Pascal's Triangle?
Pascal's Triangle arranges binomial coefficients in a triangle. Each entry is C(n,r) where n is the row and r is the position. Each number is the sum of the two numbers above it. Row 4: 1, 4, 6, 4, 1 = C(4,0) through C(4,4).
Related Tools
