🎲Permutations & Combinations Calculator

This permutations and combinations calculator computes nPr (permutations) and nCr (combinations) for any values of n and r. The central question in every counting problem is whether order matters. When it does, count permutations. When it does not, count combinations. Getting this distinction right is the key to solving probability, statistics, and combinatorics problems accurately.

How to Calculate Permutations vs. Combinations

Both formulas use the factorial function. The factorial of n (written n!) is the product of all positive integers from 1 to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120. Factorials grow rapidly: 10! = 3,628,800 and 20! is approximately 2.4 x 10^18.

• Permutations (nPr): P(n, r) = n! / (n - r)!. Use this when you are choosing r items from n and the order of the chosen items matters. Selecting the same items in a different sequence counts as a different result.
• Combinations (nCr): C(n, r) = n! / (r! x (n - r)!). Use this when you are choosing r items from n and the order does not matter. Any arrangement of the same selected items counts as one result.

The relationship between them is: P(n, r) = C(n, r) x r!. Every combination of r items can be arranged in r! different orders, so multiplying the combination count by r! gives the permutation count.

What Are Permutations? When Order Matters

A permutation is an ordered arrangement. P(n, r) counts all distinct ordered sequences of r items drawn from a pool of n. If you have 8 runners in a race and want to assign gold, silver, and bronze medals, each possible podium is a different permutation even if the same three runners appear. P(8, 3) = 8! / 5! = 8 x 7 x 6 = 336 possible medal orderings.

Order matters whenever position, rank, or sequence makes two outcomes distinct:

• Creating a PIN or password (1234 and 4321 are different codes).
• Assigning ranked prizes: 1st, 2nd, and 3rd place represent different outcomes for the same three finishers.
• Scheduling ordered tasks where sequence changes the result.
• Determining the batting order in a sports lineup.

Permutation Calculator with Repetition

The standard formula P(n, r) = n! / (n - r)! assumes no item is reused. When repetition is allowed (for example, each digit in a PIN can appear multiple times), the count is simply n^r. A 4-digit PIN from digits 0 through 9 with repetition allowed has 10^4 = 10,000 possibilities. Without repetition it would be P(10, 4) = 10 x 9 x 8 x 7 = 5,040.

What Are Combinations? When Order Does Not Matter

A combination is an unordered selection. C(n, r) counts the number of distinct groups of r items you can choose from n, where rearranging the chosen items does not create a new outcome. Selecting a 5-card poker hand from 52 cards is a combination problem: the hand {A, K, Q, J, 10} is the same hand regardless of the order the cards were dealt. C(52, 5) = 2,598,960 possible distinct hands.

Order does not matter when membership in a group is what you are counting:

• Choosing a committee or a team (no assigned roles).
• Selecting pizza toppings (the pizza is the same regardless of which topping goes on first).
• Picking lottery numbers where the winning condition depends only on which numbers were drawn.
• Choosing which questions to answer on an exam.

Combination Calculator for Lottery Numbers

Lottery problems are a classic application of combinations because winning depends entirely on which numbers match, not the order they were drawn. For a lottery that asks you to pick 6 numbers from 1 to 49, the total number of possible tickets is C(49, 6) = 13,983,816. The probability of holding the single winning ticket is 1 in 13,983,816, roughly 0.0000072 percent. Adding a bonus ball drawn from the remaining 43 numbers multiplies the sample space by 43, reaching over 600 million possibilities in some formats.

How Permutations and Combinations Apply to Probability

Probability calculations frequently use these counting techniques to determine the size of the sample space (all possible outcomes) and the number of favorable outcomes. The probability of an event equals the number of favorable outcomes divided by the total number of equally likely outcomes.

Example: what is the probability of being dealt a three-of-a-kind in a 5-card poker hand? Favorable outcomes: choose 1 rank from 13 for the triple (C(13,1)), choose 3 suits from 4 (C(4,3)), choose 2 different ranks from the remaining 12 for the pair of singletons (C(12,2)), and choose 1 suit for each singleton (4 x 4). Total favorable = 13 x 4 x 66 x 16 = 54,912. Total possible hands = C(52,5) = 2,598,960. Probability = 54,912 / 2,598,960 approximately 2.11 percent.

Pascal's Triangle and the Counting Principle

Combination values form Pascal's triangle, where the entry in row n at position r is C(n, r). The identity C(n, r) = C(n-1, r-1) + C(n-1, r) shows that each entry is the sum of the two directly above it. Row 5 reads: 1, 5, 10, 10, 5, 1, corresponding to C(5,0) through C(5,5). Pascal's triangle is central to the binomial theorem, which expands (a + b)^n using combination coefficients as the multipliers for each term.