What is the difference between nCr and nPr?

nPr counts ordered arrangements, while nCr counts unordered selections. nCr = nPr / r!, because each combination of r items can be arranged in r! different orders, all of which are counted separately in permutations.

What does C(n, 0) equal?

C(n, 0) = 1 for any n. There is exactly one way to choose zero items from a set — by choosing nothing. This is consistent with 0! = 1 in the formula.

What does C(n, n) equal?

C(n, n) = 1. There is exactly one way to choose all n items from a set of n items.

How are combinations used in probability?

Combinations are used to count favorable outcomes in probability problems. For example, the probability of getting exactly 3 heads in 5 coin flips uses C(5, 3) = 10 favorable outcomes out of 2^5 = 32 total outcomes.

What is Pascals triangle and how does it relate?

Each entry in Pascals triangle is a combination value. The entry in row n and position r (0-indexed) equals C(n, r). This makes Pascals triangle a visual representation of all combination values and is useful in binomial expansions.

Math

↵ to calculate

Combination Calculator - nCr Formula Combinations Online

Calculate combinations nCr — the number of ways to choose r items from n items where order does not matter.

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How to use this calculator

👉 Fill in the boxes below and your answer appears instantly — no maths needed, we do it all for you! 🎉

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In plain English — what does this do?

🔢 This tool does the maths for you! Just type in your numbers, and it gives you the answer right away. No need to count on your fingers or use a pen and paper.

📌 Direct Answer & Summary

A combination calculator computes nCr — the number of ways to choose r items from n items where order does not matter. Combinations are used in probability, statistics, and combinatorics. Unlike permutations, selecting items A, B, C and C, B, A count as the same combination because the arrangement is irrelevant.

n (total items)

r (items chosen)

What is Combination Calculator - nCr Formula Combinations Online?

How to use it

1️⃣ Enter n, the total number of items available.
2️⃣ Enter r, the number of items to choose.
3️⃣ The main result shows nCr with the formula used.
4️⃣ The cards below display nPr (the related permutation count) and r! for reference.
5️⃣ r must not exceed n — an error message will appear if it does.

Formula

C(n, r) = n! / (r! x (n - r)!)

💡 See it in action — a real example

How many ways can a committee of 3 be chosen from 10 candidates? C(10, 3) = 10! / (3! x 7!) = 720 / 6 = 120. There are 120 possible committees. Note that C(10, 3) = P(10, 3) / 3! because combinations remove the ordering that permutations count.

❓ Common questions

What is the difference between nCr and nPr?: nPr counts ordered arrangements, while nCr counts unordered selections. nCr = nPr / r!, because each combination of r items can be arranged in r! different orders, all of which are counted separately in permutations.
What does C(n, 0) equal?: C(n, 0) = 1 for any n. There is exactly one way to choose zero items from a set — by choosing nothing. This is consistent with 0! = 1 in the formula.
What does C(n, n) equal?: C(n, n) = 1. There is exactly one way to choose all n items from a set of n items.
How are combinations used in probability?: Combinations are used to count favorable outcomes in probability problems. For example, the probability of getting exactly 3 heads in 5 coin flips uses C(5, 3) = 10 favorable outcomes out of 2^5 = 32 total outcomes.
What is Pascals triangle and how does it relate?: Each entry in Pascals triangle is a combination value. The entry in row n and position r (0-indexed) equals C(n, r). This makes Pascals triangle a visual representation of all combination values and is useful in binomial expansions.

⚡ Related Math Tools & Calculators

Permutation Calculator Probability Calculator Mean Median Mode Calculator Standard Deviation