Resources · Probability & Statistics
Combination Formula
Combinations are essential for calculating odds, probabilities, and card hands. Understanding how to evaluate binomial coefficients simplifies statistics problem-solving. Solve selections instantly with our combination calculator or check ordered selections using the permutation calculator.
7 min read
What is the combination formula?
The combination formula (often written as C(n, r) or nCr) calculates the number of unique subsets of size r that can be formed from a total pool of n distinct elements:
C(n, r) = n! / (r! × (n − r)!)
Variables Explained
- C(n, r) = Number of combinations (possible groupings).
- n = Total number of items in the pool.
- r = Number of items being selected from the pool (where r is less than or equal to n).
- ! = Factorial operator (multiplying descending integers).
Worked Examples
Example 1: Choosing a Committee
A manager wants to choose a committee of 3 employees from a team of 5. How many unique committees can be formed?
- Formula: C(n, r) = n! / (r! × (n − r)!)
- Given: n = 5, r = 3
- Calculation: C(5, 3) = 5! / (3! × (5 − 3)!) = 5! / (3! × 2!)
- Expand: (5 × 4 × 3!) / (3! × 2 × 1) = (5 × 4) / 2 = 20 / 2
- Result: C(5, 3) = 10 ways
Example 2: Selecting Lottery Numbers
In a small game, you choose 2 numbers from 6. How many combinations are possible?
- Formula: C(n, r) = n! / (r! × (n − r)!)
- Given: n = 6, r = 2
- Calculation: C(6, 2) = 6! / (2! × (6 − 2)!) = 6! / (2! × 4!)
- Expand: (6 × 5 × 4!) / (2 × 1 × 4!) = 30 / 2
- Result: C(6, 2) = 15 ways
Common Combination Calculation Mistakes
- Using Permutations Instead: Using P(n, r) = n! / (n - r)! when order does not matter. This will fail to divide by r!, giving a count that is too high because it treats different arrangements of the same items as distinct.
- Swapping N and R: Remember that n is the total pool size and r is the sample size. You cannot choose a sample larger than the total pool (r must always be less than or equal to n).
- Factorial Expansion Errors: Be careful not to subtract terms before computing factorials in the denominator. For example, (5 - 3)! is 2! = 2, not 5! - 3! = 120 - 6 = 114.
Run the numbers
Solve ordered or unordered statistical groupings instantly:
Frequently asked questions
A combination is a selection of items from a larger set where the order of selection does not matter. For example, selecting three students for a committee is a combination because the order they are chosen doesn't change the committee.
In a combination, order does not matter (e.g., choosing 3 fruit toppings for yogurt). In a permutation, order does matter (e.g., creating a 3-digit lock code or determining 1st, 2nd, and 3rd place in a race).
The exclamation point represents a factorial. A factorial is the product of all positive integers less than or equal to that number. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1.
Both C(n, n) and C(n, 0) are always equal to 1. There is only exactly 1 way to select all n items or 0 items from a pool of n.