The formula for combinations helps us calculate how many ways we can choose a group of items from a larger set when the order does not matter. This concept appears in probability, statistics, counting problems, and almost every competitive exam. Whether you’re choosing a team, selecting lottery numbers, or solving a maths question, combinations tell you the total number of possible selections. The formula may look complex at first, but once you understand factorials and the logic behind removing order, it becomes very simple to use.
Formula for Calculating Combinations Overview
| Formula | Variables & Meaning | When It Is Used |
|---|---|---|
| nCr = n! / (r! × (n – r)!) | n = total items, r = chosen items, ! = factorial | When the order does not matter (teams, committees, lottery, probability) |
What is a Combination in Maths?
A combination is a method of selecting r items from n items when the arrangement or order of selection does not matter. It represents the number of unique groups you can form from a larger set. The formula
nCr = n! / (r! × (n – r)!)
comes from the idea that first you count all possible arrangements (n!), then divide by r! to remove the internal order of the chosen set, and divide by (n – r)! to remove the order of the unchosen items.
A few useful facts:
- nCr = nC(n – r) → choosing r items is the same as leaving n – r items.
- If r > n, then nCr = 0 (you can’t choose more items than available).
- Combinations appear in probability, card problems, selection questions, and competitive exams like CUET, SSC, JEE, Banking, etc. They also apply to day-to-day scenarios like forming sports teams or planning menu choices.
Examples to Calculate Combinations
Example 1: Choose 2 students from 5 students
Step 1: n = 5, r = 2
Step 2: 5C2 = 5! / (2! × 3!)
= 120 / (2 × 6)
= 120 / 12
Result: 10 ways
There are 10 possible ways to choose 2 students out of 5.
Example 2: Choose 3 books from 7 books
Step 1: n = 7, r = 3
Step 2: 7C3 = 7! / (3! × 4!)
= 5040 / (6 × 24)
= 5040 / 144
Result: 35 ways
There are 35 possible ways to choose 3 books out of 7.
FAQs about Combination Formula
Q1. When do we use combinations instead of permutations?
Use combinations when order does not matter. Use permutations when order matters.
Q2. Is 0C0 defined?
Yes. 0C0 = 1, meaning there is exactly one way to choose nothing from nothing.
Q3. Why do we divide by r! in the formula?
Because r! removes the internal order of the selected items, which shouldn’t affect combinations.
Q4. Can combinations be used when items repeat?
Standard combinations assume distinct items. For repeated items, use “combinations with repetition” formulas.
Q5. Are combinations important for exams?
Yes, they frequently appear in CUET, SSC, Banking, JEE, and aptitude sections.
Q6. How can we compute large nCr values quickly?
Use symmetry (nCr = nC(n – r)), cancel factors early, or use calculators / binomial tools.