Calculate C(n, r) combinations without repetition. Full enumeration for small cases, Pascal's triangle row, probability of a specific combo, and reference table.
Combinations without repetition count the number of ways to select r items from n distinct items when order doesn't matter and each item can be chosen at most once. The formula C(n, r) = n! / (r!(n−r)!) is also called the binomial coefficient and is fundamental to probability, statistics, and discrete mathematics.
This calculator computes C(n, r) for any valid input, enumerates all combinations for small cases, shows the corresponding Pascal's triangle row, and provides the probability of any single combination. It also compares with ordered permutations P(n, r) to illustrate the role of r! in eliminating order.
Applications span lottery odds (choosing 6 of 49 numbers), poker hand probabilities (5 of 52 cards), committee selection, sample subsets, and any scenario where you pick items from a pool without caring about the arrangement. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case.
While C(n,r) is simple in theory, the factorial calculations overflow quickly and mental math fails for moderate n. This calculator handles large values, shows the full enumeration when feasible, and provides the Pascal's triangle context and reference table. The visual bar chart of C(n, k) for different k illustrates where the maximum falls.
Combinations: C(n, r) = n! / (r! × (n−r)!) Permutations: P(n, r) = n! / (n−r)! Relationship: C(n, r) = P(n, r) / r! Symmetry: C(n, r) = C(n, n−r) Pascal's Rule: C(n, r) = C(n−1, r−1) + C(n−1, r)
Result: C(49, 6) = 13,983,816
A lottery requiring 6 correct numbers from 49 has 13,983,816 possible combinations. The probability of matching all 6 is 1/13,983,816 ≈ 7.15×10⁻⁸. If order mattered it would be 49!/43! = 10,068,347,520 — the r!=720 factor accounts for discarding order.
C(n,r) is the backbone of discrete probability. The probability of exactly k successes in n independent trials (each with probability p) is C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ — the binomial distribution. It appears in sampling (choosing a committee), quality control (defective items in a batch), and genetics (allele combinations).
Direct factorial computation overflows quickly (70! > 10¹⁰⁰). Use iterative multiplication: C(n,r) = ∏(i=0 to r−1) (n−i)/(i+1), computing each multiplication and division in sequence. For very large values, use logarithms: log C(n,r) = Σ log(n−i) − Σ log(i+1). Many programming languages provide built-in functions.
Binomial coefficients appear in: the binomial theorem, Pascal's triangle, Catalan numbers, Fibonacci identities, combinatorial proofs, graph theory (counting subgraphs), coding theory (error-correcting codes), and the lattice path counting problem (paths on a grid).
Combinations ignore order (choosing {1,2,3} = {3,1,2}). Permutations consider order (1-2-3 ≠ 3-1-2). C(n,r) = P(n,r)/r! — the r! accounts for the different orderings of the same r items.
The binomial coefficient, "n choose r," or the combination number. It appears in the binomial theorem: (a+b)ⁿ = Σ C(n,k) × aⁿ⁻ᵏbᵏ, giving each term's coefficient.
A triangular array where each entry is the sum of the two entries above it. Row n gives all C(n,k) for k = 0 to n. It reveals symmetry (C(n,k) = C(n,n−k)), recursion (Pascal's rule), and many other combinatorial identities.
For a 6/49 lottery, the odds are 1 in C(49,6) = 1 in 13,983,816. For Powerball (5/69 + 1/26), it's 1 in C(69,5)×26 = 1 in 292,201,338. Each combination has equal probability.
C(n, r) involves factorials, which grow faster than exponential functions. C(100, 50) ≈ 10²⁹. However, for fixed r, C(n,r) is a polynomial of degree r in n, which is manageable.
When items can be selected more than once and order doesn't matter. Standard combinations: choosing 5 cards from a deck (each card unique). With repetition: choosing 3 scoops from 5 ice cream flavors (same flavor can repeat).