Permutations And Combinations: CBSE Class 11 Maths Chapter Notes

Key Terms in Permutations and Combinations

Factorial

The product of all positive integers up to a given integer 'n'. It is denoted by n!. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1.

Fundamental Principle of Counting (Multiplication)

If an event can occur in 'm' different ways, and another independent event can occur in 'n' different ways, then the total number of ways both events can occur in order is m × n.

Fundamental Principle of Counting (Addition)

If an event can occur in 'm' different ways, and another mutually exclusive event can occur in 'n' different ways, then the total number of ways either of the events can occur is m + n.

Permutation

An arrangement of a number of objects in a definite order. The order of arrangement is important. Denoted by nPr, it represents the number of permutations of 'n' things taken 'r' at a time.

Combination

A selection of a number of objects where the order of selection does not matter. Denoted by nCr, it represents the number of combinations of 'n' things taken 'r' at a time.

Circular Permutation

An arrangement of objects in a circle. The number of circular permutations of 'n' distinct objects is (n-1)!.

The Foundation: Fundamental Principle of Counting

The entire chapter of Permutations and Combinations is built upon two simple yet powerful ideas: the Multiplication Principle and the Addition Principle. Understanding when to use which is the first step to solving any counting problem.

1. Multiplication Principle (The 'AND' Rule):
This principle applies when a task consists of a sequence of operations, one performed after the other. If you have to do Job 1 AND Job 2, you multiply the number of ways to do each. Think of it as filling a series of empty slots. For example, if you have 3 shirts and 2 pairs of pants, the total number of different outfits you can create is 3 × 2 = 6. This is because for each shirt, you have two choices of pants. The task is 'Choose a shirt AND choose pants'.

2. Addition Principle (The 'OR' Rule):
This principle applies when you have a choice between two or more mutually exclusive options. This means if you choose one option, you cannot choose the other. The task is 'Do Job 1 OR Do Job 2'. For example, a student wants to travel from Delhi to Mumbai. There are 2 train routes and 3 flight routes. The student can either take a train OR take a flight. The choices are mutually exclusive (you can't be on a train and a flight at the same time). Therefore, the total number of ways to travel is 2 + 3 = 5 ways.

In complex problems, you often use both principles together. Break down the problem into smaller sub-tasks. If the sub-tasks are sequential and dependent (AND), multiply. If they represent alternative, independent choices (OR), add.

Permutation (nPr) vs. Combination (nCr): The Core Difference

Must-Remember Formulas and Properties

• Factorial: n! = n × (n-1) × ... × 1. Also, n! = n × (n-1)!. Note: 0! = 1.
• Permutation Formula: The number of arrangements of 'n' distinct objects taken 'r' at a time is nPr = n! / (n-r)!.
• Combination Formula: The number of selections of 'n' distinct objects taken 'r' at a time is nCr = n! / (r! * (n-r)!).
• Relation between nPr and nCr: nPr = r! × nCr.
• Permutations with Repetition: The number of permutations of 'n' distinct objects taken 'r' at a time where repetition is allowed is n^r.
• Permutations of Non-Distinct Objects: The number of permutations of 'n' objects where p1 objects are of one kind, p2 are of a second kind, ..., pk are of a k-th kind is n! / (p1! p2! ... * pk!). Example: Number of arrangements of the word 'MISSISSIPPI'.
• Key Combination Properties: - nC0 = 1, nCn = 1 - nCr = nC(n-r)
• Pascal's Rule: nCr + nC(r-1) = (n+1)Cr. This is useful for simplifying expressions.
• Total Combinations: The total number of ways to select some or all items from 'n' distinct items is 2^n - 1.
• Circular Permutation: The number of ways to arrange 'n' distinct objects in a circle is (n-1)!. If there is no distinction between clockwise and anti-clockwise arrangements (like in a necklace of beads), the number of ways is (n-1)! / 2.

Worked Mini-Examples

• Permutation Example How many 4-letter words, with or without meaning, can be formed from the letters of the word 'NUMBER', assuming no letter is repeated?
• Combination Example A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done?
• Non-Distinct Objects Example How many different arrangements can be made from the letters of the word 'SUCCESS'?

Board Exam Traps and Tips

Examiners often test your fundamental understanding. The most critical skill is identifying whether to use permutation or combination.

• The Golden Question: Always ask yourself, "Does the order of selection/arrangement matter?" If yes, it's a Permutation. If no, it's a Combination.
• 'AND' vs 'OR': Be careful with problems having multiple conditions. If conditions must be met simultaneously (e.g., 'select 2 men and 3 women'), use the Multiplication Principle. If there are alternative cases (e.g., 'at least 2 men' which means '2 men OR 3 men OR ...'), use the Addition Principle for the cases.
• Repetition Allowed/Not Allowed: Read the question carefully. If repetition is allowed (e.g., forming a 3-digit number from digits 1, 2, 3, 4, 5 with repetition), you use n^r. If not allowed, you use nPr.
• Dividing for Identical Items: Don't forget to divide by the factorial of the counts of repeated items (like in 'MISSISSIPPI'). This is a very common mistake.

Quick Revision Check

• Q: Calculate the value of 8P3. A: 8P3 = 8! / (8-3)! = 8! / 5! = 8 × 7 × 6 = 336.
• Q: Calculate the value of 12C9. A: Using the property nCr = nC(n-r), we get 12C9 = 12C(12-9) = 12C3. 12C3 = (12 × 11 × 10) / (3 × 2 × 1) = 2 × 11 × 10 = 220.
• Q: When forming a team of players, do you use permutations or combinations? A: Combinations, because the order in which players are selected for a team does not matter.
• Q: In how many ways can 5 people be seated around a circular table? A: This is a circular permutation. The number of ways is (n-1)! = (5-1)! = 4! = 24.

Frequently Asked Questions

Frequently Asked Questions