Math

How Many Combinations Calculator

Count All Possible Combinations

Enter the total number of items and how many you want to select. The calculator determines whether order matters and whether repetition is allowed, then computes the exact count using the appropriate formula. Step-by-step solutions help you understand the method, not just the answer.

Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Lottery Draw

Calculate the number of possible 6/49 lottery combinations.

Key values: Total balls: 49 · Drawn: 6 · Combination mode

Race Podium

Find the number of ways to award Gold, Silver, and Bronze among 8 runners.

Key values: Runners: 8 · Medals: 3 · Permutation mode

Secret Santa

Calculate valid gift assignments where nobody draws their own name.

Key values: Participants: 8 · Derangement mode

Password Keyspace

Estimate the number of possible 8-character passwords from printable ASCII.

Key values: Character set: 94 · Length: 8 · Repetition allowed

Documentation

This calculator is also known as How Many Combinations Calculator.

Read the complete guide

Choosing the Right Formula

The correct formula depends on two questions: does order matter, and is repetition allowed?

Order matters, no repetition: P(n,r) = n!/(n-r)!
Order matters, repetition: n^r
Order does not matter, no repetition: C(n,r) = n!/(r!(n-r)!)
Order does not matter, repetition: C(n+r-1, r)

Examples

Choosing 3 from 10

How many ways to choose 3 items from 10?

C(10, 3) = 120 possible selections when order does not matter.

Key takeaway: If order mattered, P(10,3) = 720 -- six times as many.

Solving Counting Problems

Approach any "how many" problem systematically:

Identify n (total available items) and r (number selected)
Determine whether the order of selection creates a different outcome
Check whether items can be reused (repetition allowed)

Frequently Asked Questions about How Many Combinations Calculator

What is the difference between combinations with and without repetition?

Without repetition: each item can be chosen at most once (e.g., lottery draws). With repetition: items can be chosen more than once (e.g., choosing ice cream scoops from flavors). The formulas are C(n,r) and C(n+r-1,r) respectively.

When should repetition be allowed in a counting problem?

Allow repetition when the same item can appear multiple times in a selection -- for example, generating passwords (same character can repeat), rolling dice (same number on multiple dice), or choosing ice cream scoops. Disallow repetition for lottery draws, committee selection, or assigning unique roles.

How do I choose between the four counting formulas?

Answer two questions: Does order matter? Can items repeat? Order + no repeat → P(n,r) = n!/(n-r)!. Order + repeat → n^r. No order + no repeat → C(n,r) = n!/(r!(n-r)!). No order + repeat → C(n+r-1,r).