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 a combination in Maths?
In mathematics, a combination refers to a selection of items from a larger set where the order of selection does not matter. It is purely about choosing a group, not arranging it. For example, if you choose 3 fruits from a basket containing an apple, a banana, and a cherry, the group {apple, banana, cherry} is just one combination, regardless of the order you picked them in.
2. What is the formula used to calculate combinations?
The formula to calculate the number of combinations when choosing 'r' items from a set of 'n' items is:
nCr = n! / [r! * (n-r)!]
Where:
- n is the total number of items available.
- r is the number of items you are choosing.
- ! denotes the factorial, which is the product of all positive integers up to that number (e.g., 4! = 4 × 3 × 2 × 1 = 24).
3. What is the main difference between a combination and a permutation?
The key difference lies in whether order matters.
- In a combination, the order of selection is irrelevant. It is about the group itself. Example: Choosing three members for a committee.
- In a permutation, the order of selection is crucial. It is about both selection and arrangement. Example: Awarding gold, silver, and bronze medals to three winners.
Essentially, a permutation is an ordered combination.
4. What are some real-life examples where combinations are used?
Combinations are used in many everyday scenarios where the order of items is not important. Examples include:
- Forming a team: Selecting 5 players for a basketball team from a group of 10.
- Choosing food: Picking any 3 toppings for a pizza from a list of 8 available options.
- Lottery numbers: Selecting 6 numbers for a lottery ticket; the order in which the numbers are drawn does not affect the win.
- Card games: Calculating the number of possible 5-card hands you can be dealt from a standard 52-card deck.
5. Why is order not important in a combination?
Order is not important in a combination because its fundamental purpose is to count the number of possible subsets or groups, not arrangements. The identity of a group is defined by its members, not the sequence in which they were chosen. For example, a committee formed with members {Amit, Priya, Rohan} is the exact same committee as {Rohan, Priya, Amit}. Since the resulting group is identical, we count it only once.
6. In which situations would a selection problem NOT be a simple combination?
A selection problem is not a simple combination if there are additional conditions or constraints, such as:
- Order matters: If positions are assigned (e.g., President, Vice President), it becomes a permutation.
- Repetition is allowed: Choosing 3 scoops of ice cream from 5 flavours where you can pick the same flavour multiple times requires the formula for combinations with repetition.
- Items are not distinct: If you are selecting from a set with identical items (e.g., arranging the letters in the word 'APPLE'), you need specific permutation formulas.
- Restrictions apply: If a particular person must be included in a committee, or two specific people cannot be together, the problem requires a multi-step approach and cannot be solved with a single nCr formula.
7. What does the property nCr = nC(n-r) mean?
The property nCr = nC(n-r) signifies that the number of ways to choose 'r' items from a set of 'n' is the same as the number of ways to reject (n-r) items from that same set. For example, selecting a team of 3 players from 10 (10C3) is mathematically equivalent to choosing the 7 players who will not be on the team (10C7). This property provides a useful shortcut in calculations, especially when 'r' is a large number.
8. How do combinations relate to the Binomial Theorem?
Combinations are the foundation of the Binomial Theorem, which describes the algebraic expansion of powers of a binomial (a + b)^n. The coefficients of each term in the expansion are given by combination values. For instance, in the expansion of (a + b)³, the coefficients are 3C0, 3C1, 3C2, and 3C3. Each coefficient nCr represents the number of ways to choose 'r' instances of 'b' (and thus 'n-r' instances of 'a') from the 'n' binomial factors.
9. How do you calculate the number of combinations if you have to choose 3 books from a set of 5?
To calculate this, you use the combination formula nCr where n=5 (total books) and r=3 (books to choose).
The calculation is:
- 5C3 = 5! / [3! * (5-3)!]
- 5C3 = 5! / (3! * 2!)
- 5C3 = (5 × 4 × 3 × 2 × 1) / [(3 × 2 × 1) * (2 × 1)]
- 5C3 = 120 / (6 * 2)
- 5C3 = 120 / 12 = 10
Therefore, there are 10 different combinations of 3 books that can be chosen from a set of 5.
10. What is the difference between combinations with and without replacement?
The difference lies in whether an item can be chosen more than once.
- Combination without replacement: This is the standard nCr formula. Once an item is chosen, it cannot be selected again. For example, when dealing cards from a deck, a card cannot be dealt twice.
- Combination with replacement: An item can be chosen multiple times. The formula for this is (n+r-1)Cr. For example, picking 3 coins from a bag containing a penny, a nickel, and a dime, where you put the coin back each time before picking again.
