What is the Difference Between Combination and Permutation?
The concept of combination in Maths plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding combinations is essential to solving problems where the order of items does not matter, such as forming teams or picking lottery numbers.
What Is Combination in Maths?
A combination is defined as a way of selecting items from a group, where the order of selection does not matter. You’ll find this concept applied in areas such as probability, statistics, and set theory. The main difference between combination and permutation is that order matters in permutation but not in combination.
Key Formula for Combination
Here’s the standard formula: \( C(n, r) = \frac{n!}{r! \cdot (n-r)!} \)
Step-by-Step Illustration
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Suppose you have 5 fruits and you want to choose 2. How many ways can you choose?
Step 1: Identify n = 5 (total items), r = 2 (items to choose).
Step 2: Use the formula: \( C(5,2) = \frac{5!}{2!3!} \)
Step 3: Calculate 5! = 120, 2! = 2, 3! = 6.
Step 4: \( C(5,2) = \frac{120}{2 \cdot 6} = \frac{120}{12} = 10 \)
Answer: 10 ways to choose 2 fruits from 5. -
Pick a team of 3 students from a class of 8:
Step 1: n = 8, r = 3
Step 2: \( C(8,3) = \frac{8!}{3!5!} \)
Step 3: 8! = 40320, 3! = 6, 5! = 120
Step 4: \( C(8,3) = \frac{40320}{6 \times 120} = \frac{40320}{720} = 56 \)
Answer: 56 ways to select 3 students from 8.
Combination vs Permutation Table
| Aspect | Combination | Permutation |
|---|---|---|
| Order | Does NOT matter | Order matters |
| Example | Selecting a team | Assigning prizes (1st, 2nd, 3rd) |
| Formula | \( C(n, r) = \frac{n!}{r!(n-r)!} \) | \( P(n, r) = \frac{n!}{(n-r)!} \) |
| Count | Always less or equal | Always more or equal |
Cross-Disciplinary Usage
Combination is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. For example, when coding, you often need to generate all possible groups from a dataset. In sports, combinations help decide team selections. Students preparing for competitive exams like JEE or NEET will see its relevance in various problem-solving contexts.
Real-Life Examples of Combination
- Choosing 2 toppings for a pizza out of 5 available (e.g., cheese and tomato is the same as tomato and cheese).
- Picking any 3 friends to form a group from your class.
- Selecting lottery numbers — the order in which numbers are picked doesn’t matter.
Speed Trick or Memory Tip
A quick way to remember the difference: Permutation means putting things in position, so order is important. For combination, only the group matters! Vedantu teachers often show this with simple classroom games.
Frequent Errors and Misunderstandings
- Confusing combination with permutation and counting arrangement instead of selection.
- Forgetting to use the factorial (!) function correctly in formulas.
- Trying to select more items than are present (n < r), which always gives zero combinations.
- Counting {A, B, C} and {C, B, A} as different — they're the same in combination!
Try These Yourself
- How many ways can you select 4 chocolates from 10?
- In how many ways can a committee of 2 boys and 2 girls be formed from 5 boys and 4 girls?
- Find the number of different teams of 3 that can be chosen from 7 students.
Relation to Other Concepts
The idea of combination connects closely with permutations and combinations, set theory, and probability. Once you master combination, it becomes much easier to solve advanced counting and probability problems in higher classes and competitive exams.
Classroom Tip
To quickly check if you need combination or permutation, ask yourself: “Is the arrangement or just the group important?” If yes, use permutation; if not, use combination. Vedantu’s classrooms use colourful cards or objects to make these ideas easy to remember.
We explored combination in Maths — from definition, formula, stepwise examples, errors to avoid, and how it links to other topics. With Vedantu, you can master combination and apply it confidently in your exams and daily life!
- Permutations and Combinations – Full topic explanation
FAQs on Combination in Maths: Meaning, Examples, Formula & Usage
1. What is the difference between permutation and combination?
Both permutation and combination involve selecting objects from a set to form subsets. The key difference lies in the order of selection: In a **combination**, the order doesn't matter; in a **permutation**, the order is crucial. A **combination** focuses solely on the selection of items, while a **permutation** considers both selection and arrangement. For example, choosing three fruits from a basket is a combination, while arranging three books on a shelf is a permutation.
2. What is a combination in mathematics and English?
In mathematics, a **combination** is a way of selecting items from a set where the order of selection does not matter. It represents the number of ways to choose a subset of items from a larger set. The formula is nCr = n! / [r! * (n-r)!]. In English, “combination” refers to the act of joining or mixing things together to form a whole, or a group or collection of things. The mathematical concept is often used to describe group selection or arrangements where order isn’t significant.
3. What is the formula for combinations?
The formula for combinations is nCr = n! / [r! * (n-r)!], where:
• 'n' represents the total number of items in the set
• 'r' represents the number of items to be selected
• '!' denotes the factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1)
4. How do you pronounce "combination"?
The pronunciation of "combination" is /ˌkɒm.bɪˈneɪ.ʃən/.
5. How are combinations used in real life?
Combinations are used extensively in everyday life, including:
• Selecting a team from a group of players
• Choosing toppings for a pizza
• Unlocking a combination lock
• Determining the number of possible lottery outcomes
• In probability calculations
6. What is the difference between nCr and nPr?
**nCr** (combinations) calculates the number of ways to choose 'r' items from a set of 'n' items, where order doesn't matter. **nPr** (permutations) calculates the number of ways to arrange 'r' items from a set of 'n' items, where order does matter. The relationship between them is: nPr = nCr * r!
7. Can combinations be used for repeated items (with replacement)?
Yes, a modified formula handles combinations with replacement. The formula for combinations with replacement is given by: (n+r-1)Cr, where 'n' is the number of items to choose from, and 'r' is the number of selections made.
8. What happens if n < r in the combination formula?
If n < r (the number of items to choose from is less than the number of items to select), the combination is zero (nCr = 0). You cannot select more items than are available.
9. What are some properties of combinations?
Key properties of combinations include:
• nC0 = 1
• nCn = 1
• nCr = nC(n-r)
• nCr + nC(r-1) = (n+1)Cr (Pascal's Identity)
10. Why doesn't order matter in combinations?
Order doesn't matter in combinations because the focus is on selecting a subset of items, not arranging them. For example, selecting {apple, banana} is the same as selecting {banana, apple} in a combination.
11. In language, is 'combination' always mathematical?
No, 'combination' in everyday English can refer to any merging or grouping of things, not just the mathematical concept. For instance, we might talk about a 'color combination' or a 'combination of flavors'.
12. Are all groupings combinations?
No. If the order of selection or specific rules (like choosing captains) are involved, it's likely a permutation or a selection process with constraints, rather than a simple combination.
