Permutations and combinations are two fundamental concepts in combinatorics, a branch of mathematics concerned with counting, arrangement, and probability. These concepts are crucial in various fields, including statistics, computer science, and physics.

Permutations (Arrangements)

• Definition: Ordering of objects.
• Key Point: Order matters.
• Example: 'ABC' has permutations like ABC, ACB, BAC, etc.

Combinations (Selections)

• Definition: Selection of objects regardless of order.
• Key Point: Order doesn't matter.
• Example: From 'ABC', combinations of 2 letters are AB, AC, BC.

Permutation Formulas

1. Permutation of n Distinct Objects

• Formula: $P(n) = n!$
• Use: Arrange n different objects.
• Example: 4 books arranged in $4!$ ways.

2. Permutation with Repetition

• Formula: $P(n, r) = \frac{n!}{(n-r)!}$
• Use: Choose r from n objects.
• Example: Arrange 3 out of 4 books using $P(4, 3)$ .

Combination Formulas

1. Combination of n Distinct Objects

• Formula: $C(n, r) = \frac{n!}{r!(n-r)!}$
• Use: Choose r from n objects, order irrelevant.
• Example: 3 books from 5 using $C(5, 3)$.

Applications

1. Arranging 'MATHS'

• Steps: Count letters (5). Use $5!$. Answer: 120 ways.

2. Forming a 3-member Committee from 7

• Steps: Use $C(7, 3) = \frac{7!}{3!(7-3)!}$. Answer: 35 ways.

Calculation Approach

• Identify Problem Type: Permutations (order matters) or combinations (order doesn't matter).
• Select Formula: Based on problem type.
• Define Variables: n (total objects), r (objects to arrange/select).
• Perform Calculations: Apply the formula and calculate.
• Interpret Results: Understand and explain outcome.

Example: Arranging 5 Books

• Problem: Find arrangements for 5 distinct books.
• Formula: Use permutation: $P(n) = n!$.
• Apply: Here, $n = 5$, calculate $5!$.
• Calculate: $5! = 5 \times 4 \times 3 \times 2 \times 1$.
• Result: 120 different arrangements possible.