Combinations – Class 11 Maths Complete Guide
🌟 Combinations – Class 11 Maths Complete Guide 🌟
📘 Introduction
In the journey of Class 11 Mathematics, one of the most interesting topics is Combinations. It is connected with arrangements, counting, and probability. Understanding this chapter helps in competitive exams like JEE, NEET, and Olympiads. 🎯
Simply put, Combination means selection. Unlike permutations where order matters, in combinations, the order of selection does not matter. For example, choosing students for a cricket team is a combination. 🏏
📖 What is Combination?
Definition: A combination is a selection of objects where order does not matter. 🔄
Example: If we choose 2 letters from A, B, C, then AB and BA are considered the same. Thus, {AB} is a single combination.
👉 Formula of Combination is:
C(n, r) = n! / [r! (n-r)!]
- n = total objects
- r = selected objects
- ! = factorial
🔑 Difference Between Permutations and Combinations
| Aspect | Permutation | Combination |
|---|---|---|
| Order | Matters | Does not matter |
| Formula | P(n,r) = n!/(n-r)! | C(n,r) = n!/r!(n-r)! |
| Example | Arranging books | Selecting books |
📊 Properties of Combinations
- C(n, 0) = 1
- C(n, n) = 1
- C(n, 1) = n
- C(n, r) = C(n, n-r)
- C(n, r+1) + C(n, r) = C(n+1, r+1) (Pascal’s Identity)
📌 Types of Combinations
1️⃣ Simple Combination
Choosing r objects out of n without order. Formula = C(n, r).
2️⃣ Combination with Repetition
If repetition is allowed, formula = C(n+r-1, r).
3️⃣ Special Combinations in Probability
Used in probability problems where order does not affect outcomes. 🎲
💡 Real-Life Examples of Combinations
- Selecting a team of players from a group 🏀
- Choosing lottery tickets 🎟️
- Selecting menu items in a restaurant 🍔
- Choosing subjects in school 📚
- DNA combinations in biology 🧬
📐 Solved Examples
Example 1
How many ways can 3 students be chosen from 5 students?
Solution: C(5,3) = 5! / (3! × 2!) = 10.
Example 2
Find number of ways to select 2 fruits from Apple, Banana, Orange.
Solution: C(3,2) = 3! / (2! × 1!) = 3.
Example 3
How many committees of 4 people can be formed from 10 persons?
Solution: C(10,4) = 210.
Example 4
If repetition is allowed, find number of ways to select 3 candies from 5 types.
Solution: Formula = C(n+r-1, r) = C(7,3) = 35.
⚡ Practice Questions
- Find number of ways to select 5 cards from a pack of 52 cards.
- How many ways can a team of 11 be chosen from 15 players?
- Find the number of ways to select 3 vowels from the word "EDUCATION".
- How many ways to select 4 marbles from 10 identical marbles?
- In how many ways can 6 students be divided into 2 groups of 3 each?
📈 Graphical Understanding
The growth of combinations is not as fast as permutations, but still increases with n.
🌍 Applications of Combinations
- Probability and statistics 📊
- Game theory and decision making 🎮
- Cryptography 🔐
- Computer algorithms 💻
- Data sampling 📂
🔍 Summary
- Combination = Selection without order.
- Formula: C(n,r) = n! / [r! (n-r)!]
- Special case: With repetition = C(n+r-1,r)
- Applications: Probability, teams, selection problems.
- Key property: C(n,r) = C(n, n-r).
🎯 Conclusion
Combinations play an important role in mathematics. They help us count selections, understand probability, and solve real-life problems. 🌟
By mastering combinations, students improve logical thinking and problem-solving skills. 🚀
📑 References
- NCERT Class 11 Mathematics Textbook
- R.D. Sharma – Mathematics for Class 11
- Mathematics by R.S. Aggarwal
- Paul’s Online Math Notes
- Wolfram MathWorld – Combinations
Comments
Post a Comment