Combination Calculator: Find nCr Instantly

This combination calculator computes the number of ways to choose r items from a set of n distinct items, where order does not matter. You enter the total number of items (n) and the number you want to pick (r), and the calculator returns the exact count of possible combinations (nCr). It also shows the formula and a worked example so you can verify the result by hand.

Formula

The number of combinations is given by the binomial coefficient:

nCr = n! / [r! × (n − r)!]

Where:

• n = total number of distinct items (a positive integer)
• r = number of items chosen (a non-negative integer, r ≤ n)
• n! = factorial of n (product of all positive integers up to n)
• r! = factorial of r
• (n − r)! = factorial of the difference

The result is always an integer, because the denominator divides the numerator exactly.

Worked Example

Using the default values (n = 10, r = 3):

1. Compute 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800
2. Compute 3! = 3 × 2 × 1 = 6
3. Compute (10 − 3)! = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040
4. Plug into the formula: 10C3 = 3,628,800 / (6 × 5,040) = 3,628,800 / 30,240 = 120

The calculator returns 120 ways. This matches the default output.

When to Use Combinations vs. Permutations

The key question is: does order matter?

• Combinations (nCr) – order does not matter. Example: picking 3 team members from 10 candidates. {Alice, Bob, Carol} is the same as {Carol, Bob, Alice}.
• Permutations (nPr) – order matters. Example: assigning 3 distinct roles (president, treasurer, secretary) from 10 candidates. {Alice, Bob, Carol} is different from {Carol, Bob, Alice}.

If you are counting possible lottery tickets, card hands, or committee memberships, use combinations. If you are arranging items on a shelf, ordering a playlist, or assigning ranked positions, use permutations. The calculator shows both results side-by-side so you can compare.

How to Read the Result

The combination result tells you the total number of distinct groups of size r you can form from n items. For example, 10C3 = 120 means there are 120 different 3-person committees possible from a pool of 10 people. Each group is counted once, regardless of the order members are listed.

Note that combinations grow quickly but slower than permutations. For n = 20 and r = 10, there are 184,756 combinations, but 670,442,572,800 permutations. The difference becomes enormous as n increases.

Common Mistakes and Misconceptions

• Swapping n and r – nCr equals nC(n−r), so 10C3 = 10C7 = 120. But if you enter r > n, the formula breaks (factorial of a negative number is undefined). The calculator prevents this by requiring r ≤ n.
• Using permutations when you need combinations – if order doesn’t matter, using nPr will give a much larger number. Always ask: would swapping two chosen items produce a different outcome? If no, use combinations.
• Forgetting that items are distinct – this calculator assumes all n items are unique. If items are identical (e.g., 3 identical red balls), the number of distinct groups is different and requires a multiset formula.
• Rounding errors – the result is always an integer, so any decimal output indicates a mistake. This calculator rounds to 0 decimals, so you always get a whole number.

Limits of the Formula

The standard nCr formula assumes:

• All n items are distinct.
• You choose exactly r items without replacement.
• The order of selection is irrelevant.

If you allow repetition (you can pick the same item more than once), the formula changes to n+r−1Cr (combinations with repetition). If you have groups of identical items, you need a multinomial coefficient. This calculator handles the classic case only.

Also, for very large n (say n > 170), factorials overflow standard calculators. This calculator uses arbitrary-precision arithmetic internally, so it can handle n up to about 1000, but results may be extremely long numbers.

Practical Applications

Combinations appear in many everyday situations:

• Lottery odds – the number of possible ticket combinations is nCr, where n is the number of balls and r is how many you pick. For a 6/49 lottery, that’s 49C6 = 13,983,816.
• Card games – a standard 5-card poker hand count is 52C5 = 2,598,960.
• Team selection – choosing a committee, a sports team, or a study group from a larger pool.
• Probability – if each combination is equally likely, the probability of a specific group is 1 / nCr.
• Sampling – determining how many different samples of size r can be drawn from a population of n.

Whenever you hear “choose� or “pick� without order, think combinations.

Frequently Asked Questions

What is the difference between nCr and nPr?

nCr counts groups where order doesn’t matter (e.g., choosing 3 friends from 10). nPr counts arrangements where order matters (e.g., assigning 3 different roles to 3 people from 10). The permutation result is always larger by a factor of r! because each group can be ordered in r! ways.

Can r be larger than n?

No. Mathematically, nCr is defined as 0 when r > n, because you cannot choose more items than exist. The calculator enforces r ≤ n by showing an error if you try to enter a larger r.

What if I need combinations with repetition?

If you can choose the same item more than once (e.g., picking 3 flavors from 5 where repeats are allowed), the formula becomes n+r−1Cr. For n=5, r=3, that gives 7C3 = 35, versus 5C3 = 10 without repetition. This calculator does not handle that case; use a dedicated “combinations with repetition� tool.

Why does 10C3 equal 10C7?

Because choosing 3 items to keep is the same as choosing 7 items to discard. The formula nCr = nC(n−r) holds for all valid n and r. This symmetry is useful for simplifying calculations when r is close to n.

How do I calculate combinations on a scientific calculator?

Most scientific calculators have an nCr button (often labeled as “nCr� or “C�). You enter n, press the nCr key, enter r, then press equals. If your calculator lacks that button, you can compute it manually using factorials, but for large numbers it’s impractical.

What is the largest n this calculator can handle?

Because the result is an integer, the calculator can handle n up to about 1000 before the number of digits becomes unwieldy. For n=1000 and r=500, the result has about 300 digits. Most real-world problems use much smaller n.

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in n | Total items (n) | - | 10 | 0
in r | Items chosen (r) | - | 3 | 0
out comb | Combinations (nCr) | ways | ncr(n,r) | 0
out perm | Permutations (nPr) | ways | npr(n,r) | 0