Combination Formula nCr Derivation Properties and Solved Examples
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 Explained with nCr Formula
1. What is a combination in Maths?
A combination is a way of selecting items from a group where order does not matter. In combinations, arrangements like AB and BA are considered the same. It is commonly used in probability and combinatorics when counting selections such as choosing teams or committees. The key idea is selection without considering arrangement.
2. What is the formula for combination?
The formula for combination is nCr = n! / [r!(n − r)!].
- n = total number of items
- r = number of items selected
- ! = factorial (e.g., 5! = 5×4×3×2×1)
3. How do you calculate nCr step by step?
To calculate nCr, substitute values into the combination formula and simplify. Example: Find 5C2.
- Use formula: 5C2 = 5! / [2!(5−2)!]
- = 5! / (2! × 3!)
- = (5×4×3!) / (2×1×3!)
- Cancel 3! → (5×4)/(2×1) = 20/2
- = 10
4. What is the difference between permutation and combination?
The main difference is that permutation considers order while combination does not.
- Permutation (nPr): Arrangement matters (AB ≠ BA)
- Combination (nCr): Selection only (AB = BA)
- Formula relation: nPr = nCr × r!
5. What does nCr mean in Maths?
In Maths, nCr means the number of ways to choose r items from n items without considering order. It is read as “n choose r.” For example, 6C3 represents selecting 3 items from 6 items, which equals 20.
6. Can you give a real-life example of combination?
A real-life example of combination is selecting a committee from a group of people where order does not matter. For example, choosing 2 students from 4 students:
- Use formula: 4C2 = 4! / (2!2!)
- = (4×3)/(2×1)
- = 6 ways
7. What are the properties of combinations?
Combinations follow important mathematical properties used in counting and probability.
- nC0 = 1
- nCn = 1
- nCr = nC(n−r) (symmetry property)
- nC1 = n
8. Why is nC0 equal to 1?
The value nC0 = 1 because there is exactly one way to choose nothing from a set. Using the formula: nC0 = n! / [0!(n−0)!] = n! / (1 × n!) = 1. This follows from the rule that 0! = 1.
9. How is combination used in probability?
Combination is used in probability to count favorable outcomes when order does not matter. For example, the probability of selecting 2 red balls from 5 balls (3 red, 2 blue):
- Total ways = 5C2 = 10
- Favorable ways = 3C2 = 3
- Probability = 3/10
10. What are common mistakes when solving combination problems?
Common mistakes in combination problems usually involve confusing order and factorial simplification.
- Using permutation instead of combination
- Forgetting that order does not matter
- Incorrect factorial expansion
- Not applying the symmetry rule nCr = nC(n−r)
