What is the difference between combinations and permutations?

Combinations count selections where order does not matter — choosing a committee of 3 from 10 people. Permutations count arrangements where order matters — awarding 1st, 2nd, and 3rd place from 10 competitors. C(n,r) = n! / (r!(n-r)!) while P(n,r) = n! / (n-r)!.

Why is C(n,r) always less than or equal to P(n,r)?

For any selection of r items, there are r! ways to arrange those same items. So P(n,r) = C(n,r) × r!. Because r! ≥ 1, permutations are always at least as large as combinations (equal only when r = 0 or r = 1).

Can r be larger than n?

No. You cannot choose more items than exist in the set. C(n,r) and P(n,r) are both defined as 0 when r > n.

Where are combinations used in real life?

Combinations appear in lottery odds, card game hands, genetic analysis, committee selection, network path counting, and the binomial theorem in algebra. Permutations arise in scheduling, password generation, race rankings, and cryptographic key counting.

Kalkulator Kombinasi dan Permutasi

The combinations and permutations calculator helps you instantly answer one of the most common questions in probability and statistics: in how many ways can you select or arrange a group of items from a larger set? Whether you are calculating lottery odds, figuring out team selections, or working through a combinatorics problem set, this tool does the heavy lifting so you can focus on the reasoning.

What Are Combinations and Permutations?

Combinations count the number of ways to choose r items from a set of n items when the order of selection does not matter. The formula is:

C(n, r) = n! / (r! × (n − r)!)

For example, choosing 3 students from a class of 10 to form a study group is a combination problem. It does not matter whether Alice, Bob, and Carol are chosen in that order or in any other — the resulting group is the same.

Permutations count the number of ways to choose and arrange r items from a set of n items when order does matter. The formula is:

P(n, r) = n! / (n − r)!

For example, awarding first, second, and third place medals to 3 of 10 runners is a permutation problem. Placing Alice first, Bob second, and Carol third is a different outcome from placing Carol first, Alice second, and Bob third.

The key relationship between the two is: P(n, r) = C(n, r) × r!. Because each selection of r items can be arranged in r! ways, permutations always equal or exceed the corresponding combination count.

How to Use This Calculator

Enter n — the total number of items in your set (0 to 1000).
Enter r — the number of items you are selecting or arranging (must be between 0 and n).
Read the results — C(n, r) gives combinations (order irrelevant); P(n, r) gives permutations (order matters).

If r is greater than n, the result is 0 — you cannot choose more items than exist.

Examples

Example 1 — Lottery odds

A lottery draws 6 numbers from a pool of 49. How many possible winning combinations are there?

n = 49, r = 6

C(49, 6) = 49! / (6! × 43!) = 13,983,816

There are nearly 14 million possible ticket combinations, which is why lottery jackpots are so rare.

Example 2 — Race podium

In a race with 10 competitors, how many different 1st–2nd–3rd podium outcomes are possible?

n = 10, r = 3

P(10, 3) = 10! / 7! = 10 × 9 × 8 = 720

There are 720 different ordered outcomes for the top three places.

Example 3 — Committee selection

From a group of 8 employees, a company needs to select 4 for a project committee. How many ways can the committee be formed?

n = 8, r = 4

C(8, 4) = 8! / (4! × 4!) = 70 / 1 = 70

There are 70 equally valid committees that could be formed.

Common Use Cases

Probability: Combinations form the basis of calculating event probabilities. If 3 of 10 items are defective and you pick 2 at random, the probability that both are defective is C(3,2) / C(10,2) = 3/45 ≈ 6.7%.

Card games: A standard 52-card deck produces C(52,5) = 2,598,960 possible 5-card poker hands. Specific hand probabilities (flush, full house, etc.) are computed as a ratio to this total.

Genetics and biology: The binomial coefficient C(n,k) describes how many ways k alleles can be distributed among n offspring, underpinning Hardy–Weinberg equilibrium calculations.

Network design: The number of possible direct connections (edges) between n nodes in a complete graph is C(n, 2) = n(n−1)/2.

Scheduling and assignment: Permutations determine how many ways n tasks can be assigned to n workers in a one-to-one fashion (n! total assignments).

Frequently Asked Questions

Why is C(n, 0) = 1? There is exactly one way to choose nothing from a set — the empty selection. Mathematically, 0! = 1 ensures the formula n!/(0! × n!) = 1 holds consistently.

What happens when r equals n? C(n, n) = 1 and P(n, n) = n!. Choosing all items from the set can only be done one way (as a combination), but those items can be arranged in n! different orders.

Can I use this calculator for large numbers? Yes, though results can grow extremely fast. C(1000, 500) has hundreds of digits. The calculator handles up to n = 1000.

Are C(n, r) and C(n, n−r) always equal? Yes. Choosing r items to include is equivalent to choosing n−r items to exclude. This symmetry is sometimes called Pascal’s identity complement and is a useful shortcut: C(100, 98) = C(100, 2) = 4950.