Combination Calculator - n Choose r and Binomial Counts

A combination calculator is a combinatorics tool that returns the binomial coefficient C(n, r) for any n and r. Type the total number of items and the number of items you want to choose, and the combination calculator returns the exact integer n! / (r! (n - r)!) using BigInt arithmetic so even very large counts stay precise.

• • Lottery and Prize Draw Counts: Count distinct ticket sets in a 6-from-49 lottery or Powerball draw, so you can size the odds.
• • Poker Hands and Card Games: Compute the 5-card poker hands from a 52-card deck (C(52, 5) = 2,598,960) and reuse the formula for other card games.
• • Team and Committee Selection: Count project teams, juries, or committees from a larger pool when seating order is irrelevant.
• • Sampling and Quality Inspection: Calculate samples of size r from a batch of n items for acceptance sampling, audits, or inventory checks.

The result is also called the binomial coefficient, the combination, or n choose r, and is the same number that appears as a Pascal's triangle entry and as the coefficient in a binomial expansion. Use the calculator whenever the question is how many groups of r items can be pulled from n without regard to order, and switch to a permutation formula once seating, ranking, or sequence starts to matter.

When the problem also requires the ordered arrangements, the permutation and combination calculator returns both nPr and nCr side by side so the ordered and unordered counts stay in one view.

The combination calculator applies the standard n choose r formula, simplifies the multiplication so large inputs stay fast, and returns the exact integer with the factorial form so the result is easy to verify by hand.

• n: Total number of distinct items, an integer from 0 to 100.
• r: Number of items chosen, an integer from 0 to n. The calculator returns 0 when r exceeds n.
• n!, r!, (n - r)!: Factorial values; n! is 1 x 2 x ... x n, with 0! = 1.
• C(n, r): The exact integer binomial coefficient, computed with BigInt so it stays precise for inputs as large as C(100, 50).

The formula is the one most introductory statistics and probability textbooks use, and the calculator keeps every intermediate multiplication in BigInt so the value stays exact for very large inputs such as C(100, 50), where a standard JavaScript number would round to a meaningless figure.

When the with-repetition toggle is on, the formula switches to the multiset version (r + n - 1)! / (r! x (n - 1)!) so the same item can be chosen more than once, for example the 20 ways to roll a 3-dice sum of 6 from 6 faces, the 10 ways to choose 3 pizza toppings from 4 options, or the 4 multisets that pick 2 socks from a 2-color drawer.

According to Wolfram MathWorld, the n choose r formula is C(n, r) = n! / (r! (n - r)!), and the same value is also called the binomial coefficient.

According to Omni Calculator - n choose r, the C(n, r) formula is n! / (r! (n - r)!), and the multiset variant is C(n, r) = (r + n - 1)! / (r! (n - 1)!) for selections with repetition.

Because the formula is built from n!, r!, and (n - r)!, the factorial calculator is the right companion when the user wants to read off each factorial value one at a time before plugging them into the ratio.

Four ideas make C(n, r) easier to apply, and each one maps to a real input, output, or rule the calculator exposes:

These ideas prevent the most common mistakes: counting ordered lists as unordered groups, using n! / r! instead of n! / (r! (n - r)!), and forgetting that r must be at most n.

The same binomial coefficient drives the binomial probability P(X = k) = C(n, k) p^k (1 - p)^(n - k), so the binomial distribution calculator extends the n choose r result into repeated trials on the same inputs.

Follow these five steps to compute C(n, r) for any n and r with the combination calculator:

1. 1 Enter the Total Number of Items: Type the total number of distinct items, for example 12 colored balls, 52 playing cards, or 49 lottery numbers. The calculator accepts n from 0 to 100.
2. 2 Enter the Number of Items to Choose: Type r, the items to choose, for example 5 balls, 5 cards, or 6 lottery numbers. The calculator returns 0 if r is greater than n.
3. 3 Toggle Repetition if Needed: Switch on Allow Repetition when the same item can be chosen more than once, for example dice rolls or pizza topping selections.
4. 4 Read the C(n, r) Value: Check the n choose r value in the result panel. Results up to 15 digits use thousands separators; larger counts switch to scientific notation.
5. 5 Verify With the Formula and Ladder: Use the factorial form, the multiplication ladder, and the permutation count P(n, r) = C(n, r) x r! to confirm the result.

For example, with n = 12, r = 5, and repetition off, the calculator shows C(12, 5) = 792, formula = 12! / (5! x 7!) = 792, and permutation count = 95,040, matching the bag-of-balls case where 12 distinct colors and a 5-ball draw give 792 unordered sets.

Once the C(n, r) value is known, the probability calculator divides the favorable groups by the total groups to return the matching event probability on the same sample space.

Using a dedicated n choose r combination calculator has several practical advantages over computing C(n, r) by hand:

• • Exact BigInt Binomial Coefficient: The calculator uses BigInt arithmetic so the binomial coefficient is the exact integer, not a rounded floating-point approximation, even for inputs like C(100, 50).
• • Readable Factorial Form: The formula field shows the multiplication in factorial form, for example 12! / (5! x 7!) = 792, so the user can verify the calculation step by step.
• • Multiplication Ladder for Hand Checking: The simplified product (n - r + 1)...n / 1...r matches the multiplication ladder most textbooks use.
• • Permutation Count for Cross-Checking: The P(n, r) field returns C(n, r) x r!, which makes it easy to confirm the relation P(n, r) = C(n, r) x r! on the same inputs.
• • Standard and Repetition Modes: The same calculator covers both the standard n choose r problem and the multiset form (r + n - 1)! / (r! (n - 1)!).
• • Reset for the Next Problem: Press Reset to restore the default n = 12 and r = 5, useful for tutors and students working through several n choose r problems in a row.

Most introductory C(n, r) problems in a statistics or probability unit are answered in seconds once the user stops simplifying the factorial ratio by hand. The same combination count feeds straight into lottery, poker, and committee decisions.

When the experiment has more than one independent stage and each stage is its own n choose r problem, the fundamental counting principle calculator multiplies the per-stage binomial counts to size the full sample space.

A few real-world factors change how the C(n, r) formula applies and what the calculator returns:

• • The calculator counts distinct items; if some items are identical, divide by the duplicate factorials to remove overcounted permutations, which the calculator does not do automatically.
• • It does not handle dependent or restricted selections, such as a team where two members refuse to work together; restrict the option counts at the start or split the experiment into cases.

The same n! / (r! (n - r)!) value appears in the binomial probability P(X = k) = C(n, k) p^k (1 - p)^(n - k), which is why the same C(n, r) result drives both this calculator and probability problems.

According to Wikipedia - Combination, the n choose r rule is C(n, r) = n! / (r! (n - r)!), and a 5-card poker hand drawn from a 52-card deck gives C(52, 5) = 2,598,960 distinct hands.