Combinations Without Repetition Calculator - n Choose r for Lotteries, Poker, and Committees

A combinations without repetition calculator is a combinatorics tool that returns the binomial coefficient C(n, r) = n! / (r! x (n - r)!) for any n and r, where each item can be used at most once. Type the total items and the items to choose, and the calculator returns the exact integer count of unordered selections so you can size a 6-from-49 lottery, a 5-card poker hand, or a small committee from a larger pool.

• • Lottery Ticket Counts: Count distinct ticket sets in a 6-from-49 lottery or any other k-from-N draw where no number repeats, so you can size the odds and the prize pool.
• • Poker Hands and Card Draws: Compute the 5-card hands from a 52-card deck (C(52, 5) = 2,598,960) and reuse the same formula for any card game that draws without replacement.
• • Team and Committee Selection: Count project teams, juries, or committees from a larger pool when seating order is irrelevant and no person can serve twice.
• • Sampling and Quality Inspection: Calculate sample sizes for audits, acceptance sampling, and inventory checks when each item is drawn without replacement.

The C(n, r) result is also called the binomial coefficient, the k-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 without reusing any item; switch to a permutation formula once order starts to matter.

When you also want the multiset version on the same page, the general combination calculator keeps both the standard and with-repetition modes behind a single toggle.

The calculator applies the standard n choose r formula, simplifies the multiplication with the symmetry trick 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 60.
• 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 by definition.
• C(n, r): The exact integer binomial coefficient, computed with integer arithmetic so it stays precise for inputs as large as C(60, 30).

The formula is the one most introductory statistics and probability textbooks use, and integer arithmetic keeps the value exact for inputs such as C(49, 6) = 13,983,816 and C(52, 5) = 2,598,960.

The symmetry C(n, r) = C(n, n - r) picks the smaller of r and n - r for the multiplication ladder, so the number of multiplications roughly equals the smaller choice count rather than n itself.

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

According to Omni Calculator - Combinations Without Repetition, the standard n choose r expression is n! / (r! x (n - r)!) and applies whenever each item can be chosen only once.

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 the n choose r formula 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, allowing the same item to appear twice, and forgetting that r must be at most n.

When the same item is allowed to appear more than once, the combinations with repetition calculator applies the multiset form (n + r - 1 choose r) and shows the difference between the two counts side by side.

Follow these five steps to compute the n choose r count for any n and r:

1. 1 Enter the Total Number of Items: Type the total number of distinct items, for example 49 lottery numbers, 52 playing cards, or 25 students. The calculator accepts n from 0 to 60.
2. 2 Enter the Number of Items to Choose: Type r, the items to choose, for example 6 lottery numbers, 5 cards, or 4 committee members. The calculator returns 0 if r is greater than n.
3. 3 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.
4. 4 Verify With the Formula and Ladder: Use the factorial form and the ladder to recompute the count by hand. P(n, r) = C(n, r) x r! also helps cross-check.
5. 5 Reset for the Next Problem: Press Reset to restore n = 49 and r = 6, useful when working through several n choose r problems in a row.

For example, with n = 49 and r = 6, the calculator shows C(49, 6) = 13,983,816, formula = 49! / (6! x 43!), and P(n, r) = 10,068,347,520.

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 calculator has several practical advantages over computing C(n, r) by hand:

• • Exact Integer Binomial Coefficient: Integer arithmetic keeps the result exact, not a rounded floating-point approximation, even for inputs like C(60, 30).
• • Readable Factorial Form: The formula field shows 49! / (6! x 43!) = 13,983,816, 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 ladder most textbooks use and avoids n! overflow.
• • Permutation Count for Cross-Checking: The P(n, r) field returns C(n, r) x r!, which confirms P(n, r) = C(n, r) x r! on the same inputs.
• • Locked Without-Repetition Mode: The calculator only computes the standard n choose r expression, so there is no chance of accidentally counting multisets.
• • Reset for the Next Problem: Press Reset to restore n = 49 and r = 6, useful when working through several problems in a row.

The combinations without repetition formula and its with-replacement variant sit behind two separate calculators so the standard rule is never mixed up with the multiset rule. Most introductory n choose r problems are answered in seconds once the user stops simplifying the factorial ratio. The combinations without repetition result lines up with the published odds for any k-from-N draw.

When the experiment has both ordered and unordered stages and the same n and r feed into both, the permutation and combination calculator returns both nPr and nCr side by side so the counts stay in one view.

A few real-world factors change how the combinations without repetition 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 selections, which the calculator does not do automatically. It also 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 instead.

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 C(n, r) rule is n! / (r! x (n - r)!), and a 5-card poker hand from a 52-card deck gives C(52, 5) = 2,598,960 distinct hands.

Because the same C(n, r) value sits inside the binomial probability P(X = k) = C(n, k) p^k (1 - p)^(n - k), the binomial distribution calculator extends the n choose r result into repeated trials on the same n and r.