Possible Combinations Calculator

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Reviewed by: David Chen, CFA

Use this Possible Combinations Calculator to quickly determine the number of distinct arrangements or selections you can make from a larger set of items. Choose between Combinations (order does not matter) and Permutations (order matters).

Possible Combinations Calculator

Result: Total Possibilities

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Possible Combinations Calculator Formula

The calculation depends on whether the order of selection matters (Permutations) or not (Combinations).

Combinations Formula $C(n, k)$

$$C(n, k) = \frac{n!}{k!(n-k)!}$$

Permutations Formula $P(n, k)$

$$P(n, k) = \frac{n!}{(n-k)!}$$

Formula Source: Wolfram MathWorld (Combinations) | Wolfram MathWorld (Permutations)

Variables

The calculator uses two key variables:

  • Total Number of Items (n): The size of the set you are drawing from.
  • Number of Items to Choose (k): The number of items selected from the total set.

Related Calculators

What is Possible Combinations Calculator?

A Combinations and Permutations Calculator is a tool rooted in combinatorics—a branch of mathematics dealing with counting, both as a means and an end in obtaining results. It solves the fundamental problem of determining how many ways a subset of items can be selected from a larger group.

The distinction between the two types is crucial: **Combinations** are used when the order of the selected items does not affect the outcome (e.g., choosing 3 toppings for a pizza). **Permutations** are used when the order of selection creates a unique outcome (e.g., arranging the top 3 finishers in a race).

Understanding these concepts is vital in fields ranging from statistics and data science to cryptography and daily decision-making where probability and possibility spaces need to be quantified.

How to Calculate Possible Combinations (Example)

Suppose you have 10 employees and need to select 3 to form a committee (order doesn’t matter, so it’s a Combination).

  1. Identify Variables: $n = 10$ (Total Employees), $k = 3$ (Chosen Employees).
  2. Apply the Combination Formula: $$C(10, 3) = \frac{10!}{3!(10-3)!}$$
  3. Simplify the Factorials: $$C(10, 3) = \frac{10!}{3!7!}$$
  4. Calculate the Result: $$C(10, 3) = \frac{3,628,800}{(6)(5,040)} = \frac{3,628,800}{30,240} = 120$$
  5. Conclusion: There are 120 unique ways to form the 3-person committee.

Frequently Asked Questions (FAQ)

What is the main difference between combinations and permutations?
The main difference is order. Combinations do not care about the order of selection (A, B is the same as B, A). Permutations do care about the order (A, B is different from B, A).
Can I use this calculator for large numbers?
This calculator uses standard JavaScript numbers, which accurately handle factorials up to $20!$ (2,432,902,008,176,640,000). For calculating factorials of numbers larger than 20, the results may lose precision, although the final combination/permutation result will often remain accurate for larger inputs.
What does the exclamation mark (!) mean in the formula?
The exclamation mark represents the factorial function. $n!$ is the product of all positive integers less than or equal to $n$. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.
Is replacement allowed in these formulas?
The standard formulas used in this calculator ($C(n, k)$ and $P(n, k)$) are for selection **without replacement**, meaning an item cannot be chosen more than once.
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