Combinations Calculator
Calculate the number of combinations C(n, r) — choosing r items from n without regard to order. See also Permutations Calculator and Factorial Calculator.
How to Calculate Combinations
A combination is a selection of items where order does not matter. To calculate C(n, r), divide n! by the product of r! and (n−r)!. For example, choosing 3 people from a group of 10 — it does not matter in what order they are chosen, only which 3 are selected. This is different from permutations, where order matters.
Combinations Formula
C(n, r) = n! / (r! × (n − r)!)
where:
n = total number of items
r = number of items chosen
n! = n factorial = n × (n-1) × ... × 1
Example
C(10, 3) = 10! / (3! × 7!)
= 3628800 / (6 × 5040)
= 3628800 / 30240
= 120
There are 120 ways to choose 3 items from 10.
Frequently Asked Questions
What is the difference between combinations and permutations?
Combinations count selections where order does not matter (e.g., choosing a committee). Permutations count arrangements where order matters (e.g., assigning 1st, 2nd, 3rd place). P(n,r) = C(n,r) × r!.
What is C(n, 0) and C(n, n)?
Both equal 1. There is exactly one way to choose nothing (empty set) and exactly one way to choose everything (the full set).
What is Pascal's Triangle?
Pascal's Triangle is a triangular array where each entry is a combination value C(n, r). Each number is the sum of the two numbers directly above it. Row n contains C(n, 0) through C(n, n).
Where are combinations used in real life?
Combinations are used in lottery probability, card games (poker hands), genetics (gene combinations), statistics (sampling), and computer science (algorithm analysis).
Solved Examples — Combinations
Example: How many 5-card poker hands can be dealt from a standard 52-card deck?
Solution:
Step 1: Order doesn't matter → use combinations
Step 2: C(52, 5) = 52! / (5! × 47!)
Step 3: = (52 × 51 × 50 × 49 × 48) / (5 × 4 × 3 × 2 × 1)
Step 4: = 311,875,200 / 120 = 2,598,960
Answer: 2,598,960 possible hands
Example: A pizza shop offers 10 toppings. How many different 3-topping pizzas can be made?
Solution:
Step 1: Order doesn't matter (pepperoni-mushroom-olive = olive-mushroom-pepperoni)
Step 2: C(10, 3) = 10! / (3! × 7!)
Step 3: = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120
Answer: 120 different pizzas
Example: A committee of 4 must be chosen from 6 men and 5 women with exactly 2 women. How many ways?
Solution:
Step 1: Choose 2 women from 5: C(5, 2) = 10
Step 2: Choose 2 men from 6: C(6, 2) = 15
Step 3: Multiply (independent choices): 10 × 15 = 150
Answer: 150 ways
Practice Questions
Try these on your own:
- Calculate C(12, 4) (Answer: 495)
- How many ways can you choose 3 books from a shelf of 8? (Answer: 56)
- In a lottery, you pick 6 numbers from 1–45. How many combinations exist? (Answer: 8,145,060)
- How many triangles can be formed from 7 points (no three collinear)? (Answer: C(7,3) = 35)
- C(n, 2) = 28. Find n. (Answer: n = 8)
- How many handshakes occur in a room of 15 people? (Answer: C(15,2) = 105)
Common Mistakes to Avoid
The #1 mistake is using permutations when combinations are needed (or vice versa). Ask yourself: "Does the ORDER of selection matter?" If choosing a committee of 3 from 10 people, order doesn't matter → use C(10,3) = 120. If assigning president, VP, and treasurer from 10 people, order matters → use P(10,3) = 720. Another common error is double-counting by forgetting that C(n, r) = C(n, n−r). Students also sometimes add combinations when they should multiply (use multiplication principle for independent steps, addition for mutually exclusive cases). Finally, remember r cannot exceed n — C(5, 7) is undefined because you cannot choose 7 items from only 5.
Key Takeaways
- Combinations: C(n, r) = n! / (r! × (n−r)!). Order does NOT matter.
- Symmetry property: C(n, r) = C(n, n−r). Choosing 3 from 10 = choosing 7 to leave out.
- Permutations = Combinations × r!. P(n,r) is always ≥ C(n,r).
- Multiplication principle: when making independent choices in sequence, multiply the counts.
- C(n, 0) = C(n, n) = 1. There is exactly one way to choose nothing or everything.
- Applications: probability, lottery, poker hands, sampling, genetics, and committee formation.