Combinations without Repetition Calculator

About the Combinations without Repetition Calculator

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. The 6-from-49 example is a useful stress test because the count is large while the counting rule stays simple.

When This Page Helps

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.

How to Use the Inputs

1. Enter n (total pool size) and r (items to choose).
2. Or click a preset for lottery, poker, committee, etc.
3. Review C(n,r), the corresponding permutations P(n,r), and their relationship.
4. For small cases, see all enumerated combinations.
5. Examine the Pascal's triangle row to see C(n,k) for all k.
6. Use the reference table for nearby n,r values.
7. Check the probability of selecting any specific combination.

Example Calculation

Tips & Best Practices

• C(n, r) = C(n, n−r): choosing which r to include is the same as choosing n−r to exclude.
• The maximum value in a Pascal's triangle row occurs at r = n/2 (or the two middle values if n is odd).
• For large n and small r, C(n,r) ≈ nʳ/r!, which grows polynomially rather than exponentially.
• In programming, compute C(n,r) iteratively to avoid factorial overflow: multiply and divide in alternation.
• C(n,0) = C(n,n) = 1 and C(n,1) = C(n,n−1) = n.
• The sum of all C(n,k) for k = 0 to n equals 2ⁿ (total subsets of an n-element set).

The Binomial Coefficient in Probability

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).

Computational Methods

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.

Connections Across Mathematics

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).

Frequently Asked Questions

• What is the difference between combination and permutation?

  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.

• What is C(n, r) also called?

  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.

• What is Pascal's triangle?

  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.

• What are the odds of winning the lottery?

  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.

• Why does C(n, r) grow so fast?

  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 should I use combinations with repetition instead?

  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).