Possible Combinations Calculator - nCr Counts and Enumerated Sets

A possible combinations calculator is a combinatorics tool that takes a source set of n distinct objects and a sample size r, then reports the number of ways to choose r objects from the source while ignoring the order of the picks. It returns the standard n choose r count C(n, r), can switch to the combinations with repetition count C'(n, r) when the same item is allowed to appear more than once, and lists the actual combinations as readable sets.

• • Counting k-letter codes from an alphabet: Compute how many 5-letter codes can be built from the 26 letters of the English alphabet and see the enumerated sets in the same panel.
• • Listing combinations for lottery and prize draws: Generate every 6-number combination from a fixed pool so a game designer can see the full set instead of trusting the count alone.
• • Choosing teams, committees, or project groups: Pick r people from a class, department, or club and list every possible group in alphabetical order for scheduling or fair selection.
• • Building samples in statistics homework: Count equally likely outcomes for sample spaces, hypergeometric distributions, and combinatorics problems.

The result panel always shows three numbers side by side. The combinations (no repetition) row is the standard n choose r count, the combinations (with repetition) row is the same count when the same item can be picked more than once, and the listed combinations row tells you how many of those sets are actually rendered in the enumerated list below.

Readers who already know the n choose r count and want to attach a success probability p to each set can move to the Binomial Distribution Calculator and read the binomial probability for the same n and r.

The calculator applies the two textbook combination formulas and prints both counts so a reader can see the difference without toggling anything. The n choose r count uses the factorial definition, while the combinations with repetition count uses the stars-and-bars identity C'(n, r) = C(n + r - 1, r).

• n (total objects): Number of distinct items in the source set.
• r (sample size): Number of items chosen in each combination. Must be a non-negative integer.
• C(n, r) - n choose r: Number of ways to pick r items from n items without repetition and without regard to order.
• C'(n, r) - combinations with repetition: Number of ways to pick r items from n types when the same item can be chosen more than once. Equals C(n + r - 1, r).

Both formulas rely on factorials, so the calculator computes C(n, r) using the iterative product C(n, r) = n x (n - 1) x ... x (n - r + 1) / r!. That avoids the overflow of computing large factorials directly and stays inside the safe-integer range for n up to 60 with r up to 10. The with-repetition case is built on top of the same routine: C'(n, r) is rewritten as C(n + r - 1, r), so the calculator reuses the no-repetition function with the new arguments.

According to Wikipedia, "Combination", C(n, k) = n! / (k!(n - k)!) counts the number of ways to choose k elements from a set of n elements with order ignored.

Readers who need the order-sensitive version of the same problem can open the Permutation and Combination Calculator and use the permutation formula P(n, r) = n! / (n - r)! next to the combination count.

Four short definitions keep the two formulas honest. They are written so a reader who has never opened a combinatorics textbook can still understand which formula to reach for.

Readers who want to convert the n choose r count into an event probability can open the Probability Calculator and enter the same n and r there to read equally likely outcomes for an experiment.

Type n, type r, choose whether the same item can repeat, and read the result panel. The list updates as you type, so you can copy a row straight into a homework answer without retyping the values.

1. 1 Enter the total number of objects n: Type the size of the source set in the Total number of objects n box. Use 26 for the English alphabet, 6 for a die, 10 for the digits 0-9, or any integer from 1 to 60.
2. 2 Enter the sample size r: Type the number of items in each combination. The calculator accepts r = 0 (the empty selection) up to r = 10 to keep the list readable.
3. 3 Decide whether repetition is allowed: Leave the dropdown on No for the standard C(n, r) case. Switch to Yes for the combinations with repetition C'(n, r) case.
4. 4 Optionally supply custom labels: Paste a comma- or newline-separated list such as red,green,blue to use real names in the enumerated combinations.
5. 5 Read the three result rows: The first row is C(n, r) without repetition. The second is C'(n, r) with repetition. The third is the number of sets rendered in the enumerated list.
6. 6 Copy the enumerated list: The list updates in real time. Scan for the set you need, or copy the first 300 rows for a worksheet.

A reader is preparing a probability problem about 5-letter codes. They enter n = 26 and r = 5, leave repetition off, and read 65,780 combinations without repetition. They paste the first 30 rows into a spreadsheet to double-check a homework answer, then flip the dropdown to Yes and read 142,506 combinations with repetition.

Readers who want to test whether the frequencies they get from enumerating combinations match an expected distribution can move to the Chi-Square Calculator and run a goodness-of-fit test on the same counts.

A short list of what the tool does well, and what it is not designed to do, helps you put the result in the right place in your day.

• • Two combination counts in one panel: The result panel always shows the no-repetition and with-repetition counts side by side, so you can compare them without toggling the dropdown.
• • Enumerated list of every combination: Beyond the count, the calculator prints the actual sets in lexicographic order, which is what most homework, lottery, and team-selection problems need next to the count.
• • Custom labels for real source sets: Paste the real names of the items (red, green, blue, 1, 2, 3) and the enumerated list will use those names instead of the default A, B, C, ...
• • Safe handling of large counts: The combination routine uses an iterative product instead of factorial calls, so it stays inside the safe-integer range for n up to 60 and r up to 10.
• • Worked example with the alphabet: The 26-letter, 5-letter case sits in the worked example so a reader can verify the formula by hand before trusting the panel.

Readers who want to compare a single combination count against the rest of a distribution can pair the n choose r output with the Z-Score Calculator and read how unusual that count is in standard-deviation units.

The combination count depends on n, r, and the repetition setting. A few related inputs and overflow limits change the experience without changing the count itself.

• • The enumerated list is capped at 300 rows to keep the page responsive. The full count is still shown in the result panel.
• • The calculator does not model weighted, ordered, or restricted combinations. Use the permutation calculator for order-sensitive counts.
• • Counts above about 1e15 are reported as '> 1e15' rather than a precise integer because the iterative product runs out of safe-integer headroom.

According to Wikipedia, "Combination" - Number of combinations with repetition, the combinations with repetition count uses the stars-and-bars identity and equals C(n + k - 1, k).

Readers who use the combination count as the denominator of a probability fraction can pass the count and the favorable count into the Probability Fraction Calculator to read a simplified odds ratio.