What is a combination?

A combination is a selection of items where order doesn't matter. C(n, r) counts how many different groups of r items can be formed by choosing from n distinct items, without regard to the order chosen.

What is the formula for combinations?

C(n, r) = n! / (r! × (n - r)!). For example, C(5, 2) = 5! / (2! × 3!) = 10 — there are 10 ways to choose 2 items from a set of 5.

What's the difference between a combination and a permutation?

Combinations count selections where order doesn't matter (e.g., choosing a 3-person committee). Permutations count arrangements where order does matter (e.g., 1st, 2nd, 3rd place). Use the Permutation Calculator when order matters.

Why does C(n, r) equal C(n, n - r)?

Choosing r items to include is mathematically equivalent to choosing n - r items to exclude, so the two counts are always identical. This symmetry is exactly why C(8, 3) = C(8, 5) = 56.

What happens if r is greater than n?

C(n, r) is undefined when r > n, since you can't choose more items than exist in the set. The calculator will show an error in this case.

What is C(n, 0) and C(n, n)?

Both always equal 1 — there's exactly one way to choose zero items (choose nothing) and exactly one way to choose all n items (choose everything).

How does this relate to Pascal's Triangle?

Each entry in Pascal's Triangle at row n, position r, is exactly C(n, r). The triangle is a visual table of every combination value, and it's why the coefficients of a binomial expansion (a + b)ⁿ match combination values.

Where are combinations used in real life?

Combinations are used to calculate lottery odds, poker hand probabilities, committee and team selections, and binomial probability distributions in statistics.

Combination Calculator — Find C(n, r) With Step-by-Step Solutions

Our Combination Calculator finds C(n, r), the number of ways to choose r items from a set of n distinct items when order doesn't matter. Enter n and r to get the exact result instantly, along with a step-by-step formula breakdown and a Pascal's Triangle visualization showing exactly where your combination sits among its neighbors — ideal for probability, statistics, and combinatorics problems.

Quick Answer

A combination counts unordered selections: C(n, r) = n! / (r!(n - r)!) gives the number of ways to choose r items from n distinct items where order doesn't matter. Enter n and r below for an instant result with step-by-step working and a Pascal's Triangle view.

Total items (n)

Items chosen (r)

Enter n and r, then click Calculate.

How to Use the Combination Calculator — Calculate C(n,r) Online

1
Enter the total number of items, n, into the first field.
2
Enter the number of items being chosen, r, into the second field.
3
Click 'Calculate' to find C(n, r) — the number of ways to choose r items from n total, order not counted.
4
Review the step-by-step breakdown showing the formula n! / (r!(n - r)!) applied to your values.
5
View the Pascal's Triangle visualization to see how your result relates to nearby combinations.

Why Use Combination Calculator — Calculate C(n,r) Online?

Combinations answer 'how many different groups can I form?' — the foundation of probability calculations, statistics (binomial coefficients), and everyday questions like lottery odds or committee selections. Because C(n, r) always equals C(n, n - r), there's a shortcut that avoids computing huge factorials directly, which is exactly what our calculator uses internally to stay fast and exact even for larger inputs. The Pascal's Triangle view also makes the deep connection between combinations and binomial expansion visible at a glance.

Permutation vs Combination

	Permutation P(n,r)	Combination C(n,r)
Does order matter?	Yes — different orders count separately	No — only the group of items counts
Formula	n! / (n − r)!	n! / (r!(n − r)!)
Example: n=5, r=2	P(5,2) = 20	C(5,2) = 10
Typical use case	Race rankings, PIN codes, seating order	Lottery numbers, committees, card hands

Frequently Asked Questions

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