105 LearnersLast updated on June 12th, 2025
Combinations
Combinations, also called ‘selection’ is a method we use to select items from a given set of items where the order of selection does not matter. Combinations are different from arrangements or permutations where the order of selection does matter.
What are Combinations?
Combinations are selections that are made by taking a few or all objects, regardless of their arrangements. In math, the combination means “selection of things” where we do not intend to arrange the items, we intend to only select them. For example, lottery numbers are combinations because the order in which the numbers are drawn does not matter. As long as the numbers are present, the order does not matter. This gives a huge number of possible combinations, which is why winning a lottery is difficult.
Struggling with Math?
Get 1:1 Coaching to Boost Grades Fast !
What is the formula for Combinations?
Now that we know what combinations are, let us learn about the formula that we use to easily find the number of possible combinations of objects. The formula to calculate the number of combinations is:
C(n,r) = n! / r!(n-r)!
Where:
- n is the total number of items
- r is the number of items selected (r ≤ n)
- ! (factorial) is the multiplying of a number by all positive integers below it.
Let us take an example,
If there are 5 different fruits and you want to pick 2, the number of ways you can select is
C(n,r) = n! / r!(n-r)! = C(5,2) = 5! / 2!(5-2)! = 5! / 2!3! = 5 × 4 × 3 × 2 × 1 / (2 × 1)(3 × 2 × 1) = 10
So there are different ways to choose 2 fruits from a total number of 5 fruits.
What is the difference between Permutations and Combinations?
When learning about combinations, it is important to know whether the order matters or not. To understand this distinction we need to understand the difference between permutations and combinations as it can be quite confusing to know when to use permutations or combinations.
| Permutation | Combination |
| A permutation is an arrangement of objects in a specific order. Here is the order of the objects matters. | Combination is a selection of objects in any order. In combinations, the order of the objects does not matter. |
| P(n,r) = n! / (n-r)! | C(n,r) = n! / r!(n-r)! |
| We use permutations in ranking, seating arrangements or even creating our passwords. | Combinations are used in lotteries, forming a team. |
| Example: If there are 10 contestants and 3 are chosen for 1st, 2nd, and 3rd place ranking: P(10,3) = 10! / (10-3)! =720 |
Example: Selecting any 3 winners from 10 contestants without any ranking: C(10,3) = 10 / !3!(10-3)! = 120 |
Real-life applications on Combinations
Combinations are used widely in our daily lives. Here are a few real-world applications of combinations.
Lottery draws
One of the most common uses of combinations, in lotteries a set of numbers is selected. The order in which it is drawn does not matter, this makes it a combination.
Forming sports teams
When selecting players for a team, the order in which they are chosen does not matter, what matters is who is selected.
Card games
When drawing cards for a game, the order of the cards does not matter. What matters is the cards you have. This makes it a combination.
Common Mistakes and How to Avoid Them in Combinations
When learning about combinations, students might often make mistakes. So here are a few common mistakes and how to avoid them:
Mistake 1
Getting the formula for combinations confused with permutations
A mistake that students might make is using the permutation formula confused with the combinations' formula. Students must memorize the combination formula and understand the difference between both formulas.
The combination formula is: C(n,r) = n! / r!(n-r)!.
Mistake 2
Confusing n and r in the formula.
When using the formula, students must remember what ‘n’ and ‘r’ denote. In the formula, ‘n’ is the total number of items, and ‘r’ is the number of items selected.
Mistake 3
Incorrect calculations of factorials
Solving for factorials can be quite difficult. To make it easy, students can write out the factorial expansion step-by-step and cancel the common terms. After solving, make sure to double-check the calculations.
Mistake 4
Ignoring the domain of r
Students might try to solve using combinations, even if r > 0. Make sure that r is within the valid range relative to n.
Mistake 5
Using approximate values for factorials
Estimating factorial values instead of calculating them might lead to incorrect answers. Always calculate factorials exactly or simplify the expression before any kind of calculation.
Level Up with a Math Certification!
2X Faster Learning (Grades 1-12)
Solved examples on Combinations
Problem 1
How many ways can you choose 4 books out of 8?
There are 70 combinations to choose 4 books out of 8.
Explanation
C(n,r) = n! / r!(n-r)!
C(8,4) = 8! / 4!(8-4)! = 8! / 4!4! = 40320 / 24 × 24 = 70
Problem 2
How many combinations are there if we choose 0 items from 15?
1
Explanation
C(15,0) = 1. This is because 0! = 1.
Problem 3
Out of 12 applicants, a manager needs to form a team of 4 employees. How many different teams can be formed?
There are 495 combinations to form a team of 4.
Explanation
C(n,r) = n! / r!(n-r)!
C(12,4) = 12! / 4!(12-4)! = 12! / 4!8! = 495.
Problem 4
In the word Brave there are 5 distinct letters, how many unique combinations of 3 letters can be selected?
10
Explanation
Since the order does not matter when selecting letters, we will use the combination formula.
C(n,r) = n! / r!(n-r)!
C(5,3) = 5! / 3!(5-3)! = 5! / 3!2! = 20 2 = 10.
Problem 5
A chef wants to create a new dish by choosing 4 spices from a collection of 15. How many different spice blends are possible?
1365 combinations
Explanation
C(n,r) = n! / r!(n-r)!
C(15,4) = 15! / 4!(15-4)! = 15! / 4!11! = 1365
Turn your child into a math star!
#1 Math Hack Schools Won't Teach!
FAQs on Combinations
1.What are combinations?
2.How to know if a problem requires combinations or permutations?
3.Can we use combinations when items are repeated?
4.How are combinations selected if there are larger numbers?
5.What is the combination of items if r = 1?
Struggling with Math?
Get 1:1 Coaching to Boost Grades Fast !
Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
