Probability

Probability Solver

Calculate combinations and permutations instantly with professional formatting.

Combinations

120

The number of ways to choose 3 items from 10 without regard to order.

Permutations

720

The number of ways to arrange 3 items from 10 where order matters.

Understanding Combinations and Permutations

Combinations and permutations are fundamental concepts in combinatorics — the branch of mathematics that deals with counting arrangements. They form the backbone of probability theory and are used extensively in statistics, computer science, cryptography, and game theory. TheCalcPro's probability solver computes both nCr (combinations) and nPr (permutations) with step-by-step formula breakdowns so you can understand the counting logic, not just the final number.

The Formulas

Permutations: P(n, r) = n! / (n − r)!

Combinations: C(n, r) = n! / (r! × (n − r)!)

Factorial: n! = n × (n−1) × (n−2) × … × 2 × 1

Where n is the total number of items and r is the number being selected. The key difference: permutations count ordered arrangements, while combinations count unordered selections. Combinations divide by r! to remove duplicate orderings.

Step-by-Step Example: Lottery Odds

A lottery requires you to pick 6 numbers from 49 (order doesn't matter). How many possible combinations exist?

  1. Identify the formula: Since order doesn't matter, use C(49, 6)
  2. Apply the formula: C(49, 6) = 49! / (6! × 43!)
  3. Simplify: = (49 × 48 × 47 × 46 × 45 × 44) / (6 × 5 × 4 × 3 × 2 × 1)
  4. Compute numerator: 49 × 48 × 47 × 46 × 45 × 44 = 10,068,347,520
  5. Compute denominator: 720
  6. Divide: 10,068,347,520 / 720 = 13,983,816

There are 13,983,816 possible combinations, giving odds of approximately 1 in 14 million for any single ticket. This same C(n,r) formula applies to committee selection, card hand probabilities, and sampling problems in statistics. For statistical analysis of your data, explore our Statistics Engine.

Probability FAQ

Combinations (nCr) are used when the order does not matter (e.g., picking a committee of 3 from 10 people). Permutations (nPr) are used when the order does matter (e.g., choosing 1st, 2nd, and 3rd place in a race). The formula for nCr = n! / (r!(n−r)!) and nPr = n! / (n−r)!. Since nCr divides by r!, it always gives a smaller or equal result than nPr.

In mathematics, the empty product is defined as 1 by convention. This makes many formulas work consistently for edge cases. For example, nC0 = n! / (0! × n!) = 1, which correctly represents the fact that there is exactly one way to choose zero items from any set — by choosing nothing.

The calculator works for any non-negative integers where n ≥ r. For very large numbers, results may be shown in scientific notation due to JavaScript's floating-point limits (factorials beyond 170! exceed the representable range). For practical purposes, the calculator handles all common probability homework and statistical applications.

Use combinations when selecting items from a group where the arrangement doesn't matter: lottery numbers, committee selection, pizza topping choices. Use permutations when the arrangement matters: password generation, seating arrangements, race placements, phone number sequences. Ask yourself: "Does the order of selection change the outcome?" If yes, use permutations.