Permutations vs. Combinations
Permutations and combinations are fundamental concepts in combinatorics that deal with arranging and selecting objects from a set. Understanding the differences between permutations and combinations is crucial for solving a wide range of counting and probability problems.
Permutations:

Definition: Permutations are ordered arrangements of objects taken from a set without repetition. In permutations, the order of arrangement matters.

Formula: The number of permutations of "r" objects selected from a set of "n" distinct objects without repetition is calculated using the permutation formula:
 $P(n,r)=\frac{n!}{(nr)!}$

Order Matters: In permutations, the sequence of items is important. For example, arranging books on a shelf, choosing winners in a race, or creating passwords all involve permutations.

Example: If you have 4 distinct books (A, B, C, D) and you want to arrange 3 of them on a shelf, there are $P(4,3)=\frac{4!}{(43)!}=24$ different permutations.

Applications: Permutations are used in scenarios where the arrangement order is significant, such as in permutations with or without repetition in password generation.
Combinations:

Definition: Combinations are unordered selections of objects from a set without regard to the order of selection. In combinations, the order does not matter.

Formula: The number of combinations of "r" objects selected from a set of "n" distinct objects is calculated using the combination formula:
 $C(n,r)=\frac{n!}{r!(nr)!}$

Order Doesn't Matter: In combinations, the sequence of items is not important. For example, selecting a committee from a group of people, choosing items for a meal, or forming teams are all situations that involve combinations.

Example: If you have 5 distinct books (A, B, C, D, E) and you want to select 3 to read, there are $C(5,3)=\frac{5!}{3!(53)!}=10$ different combinations.

Applications: Combinations are used in situations where the arrangement order is irrelevant, such as selecting groups, teams, or subsets of items.
Key Differences:

Permutations involve order: Permutations consider the order of arrangement, making them suitable for situations where the sequence of items is essential.

Combinations are orderagnostic: Combinations do not consider the order of selection, making them suitable for situations where the order of items doesn't matter.

Different formulas: Permutations and combinations have distinct formulas reflecting their fundamental differences in counting.

Permutations are often more numerous: Permutations typically result in more possibilities than combinations, as they account for different orders.

Applications: Permutations are used in scenarios like arranging items, generating distinct passwords, or determining winners in a race. Combinations are used in scenarios like forming committees, selecting groups, or choosing items for a combination lock.