Combination Calculator
Calculate combinations (nCr) and combinations with repetition instantly with interactive Pascal's triangle, BigInt precision, and step-by-step methods.
10
Choose repetition regime and input set values
Deep-Dive Combinatorics Studio
Visualize Pascal's identity, token subsets, or order collapse
Compare all 4 fundamental counting regimes for active values n = 5 and r = 3:
Explore the mathematical mechanics substituting your inputs
The factorial of n computes all permutations, divided by r! to eliminate ordering within the selected items, and (n - r)! to eliminate ordering within unchosen items.
Audit-ready combinatorics calculation report with one-click export
======================================== COMBINATION CALCULATION RECEIPT OctaLone Tools (octalone.com/tools/combination-calculator) ======================================== Timestamp: 2026-09-09T00:00:00.000Z Mode: Without Repetition (Standard nCr) Total Items (n): 5 Items Chosen (r): 3 PRIMARY RESULT: • Combination Count: 10 • Notation: C(5, 3) = 10 • Odds: 1 in 10 • Probability: 10.0000% COMPANION COMBINATORIAL REGIMES: • Combinations Without Repetition C(n, r): 10 • Combinations With Repetition C^R(n, r): 35 • Permutations Without Repetition P(n, r): 60 • Permutations With Repetition (n^r): 125 • Order Collapse Factor (r!): 6 • Power Set Total Subsets (2^n): 32 MATHEMATICAL BREAKDOWN: • Formula: C(n, r) = n! / (r! × (n - r)!) • Variable Values: n = 5, r = 3 • Substituted: 5! / (3! × (5 - 3)!) • Symmetry Equivalent: C(5, 2) = 10 • Order Collapse Ratio: P(n, r) / C(n, r) = 6 ======================================== 100% Client-Side Privacy: No data transmitted to any external server. ========================================
While permutations consider sequence order (e.g. PIN codes 1234 ≠ 4321), combinations group items into unordered sets (hand {Ace, King} = {King, Ace}). This reduces count by exactly r! factor.
How many ways can you choose a 3-person project team from 5 engineers?
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Frequently Asked Questions
A combination is a selection of items from a collection where the order of selection does not matter. Given n items, C(n, r) (read 'n choose r') counts the number of distinct subsets of size r that can be formed without repetition.
The fundamental distinction is whether order matters. In permutations (nPr), order matters (e.g. 1234 is distinct from 4321). In combinations (nCr), order does not matter (e.g. a poker hand of Ace and King is identical to King and Ace). Every combination of size r corresponds to r! permutations.
In combinations with repetition, items may be selected more than once from n categories (for example, selecting 3 scoops of ice cream from 8 available flavors). Using the stars-and-bars method, this is equivalent to choosing r items from (n + r - 1) total slots: C^R(n, r) = C(n + r - 1, r).
Every entry in Pascal's Triangle corresponds exactly to a binomial coefficient C(n, r), where n is the row number (starting from row 0) and r is the column position. The fundamental recurrence relation C(n, r) = C(n - 1, r - 1) + C(n - 1, r) explains why each cell is the sum of the two cells directly above it.
Combinations satisfy the mathematical symmetry C(n, r) = C(n, n - r). Choosing r items to include in a group is mathematically identical to choosing the remaining (n - r) items to leave out. For example, C(10, 3) = 120 is identical to C(10, 7) = 120.
In standard lotteries like 6/49, you must match 6 unique numbers drawn from 49 balls where order does not matter. The total possible ticket combinations is C(49, 6) = 13,983,816. Therefore, the odds of winning with a single ticket are exactly 1 in 13,983,816 (about 0.00000715%).
Yes, 100%. All calculations, BigInt factorials, and interactive visualizers execute entirely client-side in your web browser. Zero numbers, parameters, or receipts are ever transmitted to any external server.