Permutations and combinations are fundamental tools in discrete mathematics, crucial for solving a wide array of problems in competitive exams like JEE. They help us count the number of ways events can occur, which is essential for probability, statistics, and various problem-solving scenarios.

Before diving into applications, let's quickly recap the definitions:

• Permutation: An arrangement of objects in a specific order. The order matters. The formula for permutations of 'n' objects taken 'r' at a time is P(n, r) = n! / (n-r)!.
• Combination: A selection of objects where the order does not matter. The formula for combinations of 'n' objects taken 'r' at a time is C(n, r) = n! / (r! * (n-r)!).

Permutations and combinations are widely applied in various types of problems encountered in competitive exams. These include:

1. Arrangement Problems: Arranging items in a line, around a circle, or in specific positions.
2. Selection Problems: Choosing a subset of items from a larger set.
3. Distribution Problems: Distributing items into distinct or identical boxes.
4. Probability Calculations: Determining the likelihood of specific events occurring.

These problems involve finding the number of ways to arrange a set of objects. For instance, arranging letters in a word, people in a queue, or books on a shelf.

• Linear Arrangements: If all objects are distinct, the number of ways to arrange 'n' objects is n!. If there are repetitions, we divide by the factorial of the counts of each repeated object.
• Circular Arrangements: The number of ways to arrange 'n' distinct objects in a circle is (n-1)!.

These problems focus on choosing a group of items from a larger set, where the order of selection doesn't matter. This is where combinations are used.

• Forming Committees/Teams: Selecting a group of people from a larger pool.
• Choosing Items: Selecting a certain number of items from a collection.
• Problems with 'At Least' or 'At Most': Often require summing up combinations for different cases.

These problems involve distributing items into boxes. The nature of the items (distinct or identical) and boxes (distinct or identical) significantly impacts the approach.

• Distinct items into distinct boxes: Each item can go into any box independently.
• Identical items into distinct boxes: Often solved using stars and bars method (combinations with repetition).
• Distinct items into identical boxes: Related to Stirling numbers of the second kind.
• Identical items into identical boxes: Related to partitions of integers.

Permutations and combinations are the backbone of many probability problems. The probability of an event is calculated as the ratio of favorable outcomes to the total possible outcomes.

• Example: The probability of drawing two red cards from a deck of 52 cards without replacement. The total number of ways to draw 2 cards is C(52, 2). The number of ways to draw 2 red cards (from 26 red cards) is C(26, 2). The probability is C(26, 2) / C(52, 2).

Mastering permutations and combinations requires careful problem analysis. Key strategies include:

• Identifying 'Order Matters' vs. 'Order Doesn't Matter': This is the most critical step.
• Breaking Down Complex Problems: Decompose problems into smaller, manageable parts.
• Using Complementary Counting: Sometimes it's easier to count the opposite event and subtract from the total.
• Handling Restrictions: Pay close attention to conditions like 'at least', 'at most', 'never', 'always'.

Practice is paramount. Work through a variety of problems from past JEE papers and reputable textbooks to build intuition and speed.