Combination Calculator

Combination Calculator – Core Combinatorics in One Place

This Combination Calculator collects the essential tools of combinatorics and puts them into a single, easy-to-use interface. Instead of switching between different calculators for nCr, nPr, factorials, combinations with repetition and probability, you can simply move between tabs and explore all of them using a consistent notation and layout.

Combinatorics is the branch of mathematics that studies how to count, arrange and select objects. It appears in probability theory, statistics, computer science, cryptography, game design, operations research and many day-to-day contexts. Whenever you ask “how many possible passwords?”, “how many ways can I choose a committee?”, or “what is the chance of this poker hand?”, you are doing combinatorics.

Combinations vs Permutations – Order Matters

At the heart of many counting problems is a single question: does order matter? The calculator separates this into two main functions: combinations and permutations.

Combinations care only about which items are selected, not the order in which they appear. Permutations care about both selection and arrangement. For example, if you choose the letters A, B and C from the set {A, B, C, D}, then the subset {A, B, C} is a combination. The arrangements ABC, ACB, BAC, BCA, CAB and CBA are six different permutations of that combination.

Combinations Formula nCr

In this formula:

• n is the total number of distinct items
• r is the number of items chosen
• n! is the factorial of n, the product 1 × 2 × 3 × … × n

The factorials in the denominator account for the fact that different orderings of the same selected items should not be counted multiple times. The calculator uses a multiplicative version of this formula internally so that it does not need to compute full factorials when n is large.

Permutations Formula nPr

For permutations, order matters. The numerator counts all ways to arrange n items. The denominator removes the trailing factors that involve items not selected. For example, the number of ways to choose a first, second and third place winner from 10 contestants is P(10, 3).

The calculator’s permutations tab uses the same multiplicative style internally, computing products like n × (n − 1) × … × (n − r + 1) directly instead of expanding full factorials.

Factorials – The Building Blocks of nCr and nPr

Factorials are the building blocks behind many combinatorics formulas. The factorial of a nonnegative integer n, written n!, is the product of all positive integers up to n:

By convention, 0! is defined to be 1. This might seem arbitrary at first, but it makes the formulas for combinations and permutations work seamlessly even when r = 0 or r = n. Factorials grow extremely quickly; 10! is already 3,628,800, and 20! is a 19-digit number.

The factorial tab in the calculator lets you explore this growth by computing n! for different values of n. Behind the scenes, it uses integer arithmetic to keep the result exact within the practical range of the tool.

Combinations With Repetition – Multisets

In many counting problems, items can be chosen more than once. For example, selecting scoops of ice cream with flavors that can repeat, choosing coins from a pile where there are many identical coins of each denomination, or selecting ingredients in a recipe. These situations are described by combinations with repetition, also known as multiset combinations.

When you have n types of items and you want to choose r items where each type can be picked multiple times, the number of possible selections is given by:

This formula arises from the stars-and-bars method. You can imagine r stars representing chosen items and n − 1 bars representing boundaries between types. The total number of positions is n + r − 1, and you choose r of them to be stars, which yields C(n + r − 1, r) possible patterns. The calculator implements this directly by computing C(n + r − 1, r) for the values you enter.

Example: Ice Cream Combinations With Repetition

Suppose there are 4 flavors of ice cream and you are ordering a cone with 3 scoops, where you allow repeated flavors. How many distinct cones are possible if the order of scoops does not matter? Using combinations with repetition:

In the calculator, you would set n = 4 and r = 3 in the repetition tab and obtain 20 as the result. This counts cones like {chocolate, vanilla, strawberry} and {chocolate, chocolate, vanilla} as distinct multisets while ignoring the physical order of scoops.

Hypergeometric Probability – Combinations in Probability Form

Combinations are also at the core of many probability calculations, especially when drawing items without replacement. The hypergeometric distribution is a classic example. It describes the probability of drawing exactly k successes when you take n draws from a finite population of size N that contains K successes and N − K failures, without replacement.

The hypergeometric probability mass function is:

Each piece has a combinatorial interpretation:

• C(K, k) counts ways to choose the k successes drawn
• C(N − K, n − k) counts ways to choose the remaining n − k draws as failures
• C(N, n) counts all possible ways to choose n items from N, regardless of type

The calculator’s hypergeometric tab computes both the top and bottom combinations and then divides them, returning both a decimal probability and a fraction string representation.

Example: Drawing Cards Without Replacement

Consider a standard deck of 52 cards with 4 aces. What is the probability that you draw exactly 2 aces in a 5-card hand? Here:

The probability is:

In the calculator, you would enter those values and obtain a numerical probability. This same pattern applies to many real-life problems: defective items in a lot, colored balls in an urn, or success/failure categories in a small population.

How the Calculator Handles Large Integers

Combinatorics expressions like nCr and nPr can grow very large, even for modest n. Instead of computing n! directly and dividing, this Combination Calculator uses multiplicative formulas that build the result step by step while canceling factors implicitly. For example, to compute C(100, 3), it can use:

This avoids handling 100! as a single gigantic intermediate value. Internal integer mathematics, often using BigInt where available, allows the calculator to keep results exact for a wide range of n and r values. At the same time, practical limits ensure that numbers remain displayable and the interface stays responsive.

Step-by-Step: Using Each Tab in the Combination Calculator

1. Combinations (nCr) Tab

1. Enter n, the total number of items in the set.
2. Enter r, the number of items you want to choose.
3. Click calculate to compute C(n, r).
4. Read the advisory note describing what the result means in terms of selections without order.

This is the tab you use for committee selection, subset counts, lottery combinations and many probability denominators.

2. Permutations (nPr) Tab

1. Enter n, the total number of distinct items.
2. Enter r, the number of items to arrange in order.
3. Click calculate to compute P(n, r).
4. Use the result to understand how many ordered outcomes are possible.

Use this for ranking problems, password arrangements without repetition, seating arrangements and ordered selections.

3. Factorial Tab

1. Enter a nonnegative integer n.
2. Click calculate to compute n!.
3. Observe how quickly factorials grow as you increase n.

This tab is perfect for exploring the growth of n! and for providing the raw data behind more complex combinatorics formulas.

4. Combinations With Repetition Tab

1. Enter n, the number of different item types available.
2. Enter r, the total number of items to select, allowing duplicates.
3. Click calculate to compute C(n + r − 1, r).
4. Interpret the result as the number of multisets of size r from n types.

This is the correct tab for problems involving selections where the same type can appear multiple times and order does not matter, such as distributions of identical objects among labeled boxes.

5. Hypergeometric Probability Tab

1. Enter N, the total population size.
2. Enter K, how many of those N items are labeled as successes.
3. Enter n, the number of draws without replacement.
4. Enter k, the number of successes you want in your sample.
5. Choose the number of decimal places for the probability.
6. Click calculate to see both the probability value and a combination-based fraction.

This tab is used for card draws, quality control sampling, and any scenario where you are drawing without replacement from a finite set with two categories.

Common Use Cases for the Combination Calculator

The advanced Combination Calculator supports many practical applications beyond pure math exercises. Some popular use cases include:

• Analyzing lotteries and raffles where combinations of numbers are drawn
• Counting the number of possible poker hands or other card game outcomes
• Designing and testing passwords and codes with or without repetition
• Estimating outcomes in randomized algorithms or simulations
• Planning experiments in statistics, such as sample sizes and treatment allocations
• Working with occupancy and distribution problems in discrete mathematics

Because the calculator gives exact counts for nCr, nPr and related functions, it can serve as a reliable reference when checking textbook answers or designing your own problems.

Understanding Limits and Interpreting Large Results

It is common for combinatorics outputs to become extremely large. Even a modest increase in n can cause nCr or nPr values to jump by several orders of magnitude. The calculator is designed to compute exact integers for a wide range, but you should interpret large results carefully. Sometimes it is more important to know the order of magnitude than the exact digits.

For probabilities, it is equally common to encounter very small values when events are rare. The hypergeometric tab lets you specify decimal places so that tiny probabilities are still visible and meaningful. When using these numbers in real-world decisions, always combine them with good judgment, domain knowledge and appropriate statistical methods.