The Fundamental Counting Principle (Multiplication Principle) states that if one event can occur in 'm' ways and a second independent event can occur in 'n' ways, then the total number of ways both events can occur together is m × n. This principle extends to any number of independent events and forms the foundation of all counting problems.

Linear Permutation deals with arranging distinct objects in a straight line (order matters). The number of ways to arrange 'n' distinct objects in a line is n! (n factorial). This fundamental concept extends to arranging only 'r' objects out of 'n' (ⁿPᵣ = n!/(n-r)!).

Combination deals with selecting objects where order does NOT matter. The number of ways to choose 'r' objects from 'n' distinct objects is denoted as ⁿCᵣ or C(n,r) = n! / [r! × (n-r)!]. Combinations are used when forming committees, selecting teams, or any situation where the arrangement of selected items is irrelevant.

Word Formation with Repetition (Permutation of Identical Objects) deals with arranging objects where some are identical. The number of distinct arrangements of 'n' objects where there are p identical objects of one type, q of another, etc., is n!/(p! × q! × ...). This is commonly applied to finding distinct anagrams of words with repeated letters.

Circular Permutation deals with arranging distinct objects around a circle (or any closed loop). Unlike linear arrangements, rotations of the same circular arrangement are considered identical because there is no fixed starting point. The number of ways to arrange 'n' distinct objects around a circle is (n-1)!.

Permutation with Restriction problems involve arranging objects with specific constraints such as: certain objects must be together, must be apart, must be at fixed positions, or must be at ends. These problems require treating restricted groups as units or using complementary counting.

Committee Formation problems involve selecting a team or committee from different groups (e.g., selecting from men and women, from different departments, or with specific skill requirements). These are combination problems with constraints that require selecting from multiple independent groups.

The Gap Method is a technique for arranging items such that certain items are not adjacent to each other. First, arrange the unrestricted items, creating gaps between them. Then, place the restricted items in these gaps to ensure separation. This method is especially useful for 'no two vowels together' or 'no two specific people together' problems.

Geometrical Combinations involve counting geometric figures (triangles, lines, diagonals, quadrilaterals, etc.) formed by joining given points in a plane. These problems combine combinatorial selection with geometric constraints, such as collinearity (points on the same line cannot form triangles).

Circular Permutation with Reflection deals with arrangements around a circle where clockwise and anticlockwise arrangements are considered identical (as in necklaces, bracelets, or garlands). Since a necklace can be flipped over, the number of distinct arrangements is (n-1)!/2 for n ≥ 3 distinct objects.

Derangement is a permutation of elements where no element appears in its original position. For example, arranging n letters into n envelopes so that no letter goes into its correct envelope. The number of derangements of n items is denoted as !n (subfactorial) or D(n).

The Inclusion-Exclusion Principle (PIE) is a counting technique used to calculate the size of the union of multiple sets when there is overlap between them. For two sets: |A ∪ B| = |A| + |B| - |A ∩ B|. For three sets: |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|.

Distribution problems involve distributing objects into boxes or recipients. When objects are distinct and boxes are distinct, each object has independent choices, so the number of ways is (number of boxes)^(number of objects). Different variations include empty boxes allowed or not, identical objects, or identical boxes.

Number Formation problems involve forming numbers (usually with a given number of digits) from a set of digits, often with restrictions such as: first digit cannot be zero, digits can/cannot repeat, or numbers must be divisible by certain values (even, odd, divisible by 5, etc.).

Group Division problems involve dividing a set of distinct items into groups (teams, batches, etc.). Key considerations: Are groups distinguishable (labeled like Team A, Team B) or indistinguishable? Are groups of equal size or different sizes? The formula involves combinations and division by factorial for indistinguishable groups.

Rank of a word is its position when all permutations of its letters are arranged in dictionary (alphabetical) order. For example, the word 'MOTHER' has a certain rank among all 6! = 720 permutations of its letters. Ranking requires systematically counting how many words come before the given word.

Selection with Mandatory Constraint problems involve selecting a committee or team where certain specific individuals must be included (or must be excluded). These problems simplify by fixing the mandatory selections first, then selecting the remaining from the available pool.

The Stars and Bars method is a combinatorial technique for counting the number of ways to distribute n identical items into k distinct boxes (or find non-negative integer solutions to x₁ + x₂ + ... + xₖ = n). The formula is C(n + k - 1, k - 1).

Combination with 'At Least' Constraint problems require selecting a committee or group where certain categories must have at least a minimum number of members (e.g., at least 2 men, at least 1 woman). These are solved by summing combinations for each valid case or using complementary counting.

Permutations with Identical Objects (Multiset Permutations) deal with arranging objects where some are identical. The number of distinct arrangements is n! / (p! × q! × r! × ...), where p, q, r are the frequencies of identical objects. This extends word formation problems to general objects.

Circular Permutations with Identical Objects (Necklace/ Garland problems with repeated beads) require counting distinct arrangements around a circle where rotations and reflections are considered identical AND some objects are identical. These are the most complex permutation problems, often requiring Burnside's Lemma.

Partial Derangement (Rencontres Numbers) counts permutations of n elements with exactly k fixed points (elements that stay in their original position). The formula is D(n,k) = C(n,k) × !(n-k), where !(n-k) is the derangement number. These generalize derangements (k=0) to allow some fixed points.

The Multinomial Theorem generalizes the Binomial Theorem to more than two terms. The coefficient of x₁ᵃ x₂ᵇ ... xₖᶜ in the expansion of (x₁ + x₂ + ... + xₖ)ⁿ is n!/(a! b! c! ...), where a + b + c + ... = n. This is also the number of ways to arrange n objects with a of type 1, b of type 2, etc.

Handshake problems count the number of handshakes that occur in a group where each pair of people shakes hands exactly once. The classic formula is C(n,2) = n(n-1)/2. Variations include constraints such as certain people not shaking hands, people shaking only within groups, or specific handshake patterns.

Partition into Unequal Groups involves dividing a set of distinct items into groups where the groups have different sizes. When group sizes are all different, the groups are naturally distinguishable by size, so no division by factorial is needed. This differs from equal groups where division by k! is required.

Number Formation with Distinct Digits involves forming numbers where digits cannot repeat. This is a permutation problem where we arrange selected digits. The number of n-digit numbers with distinct digits (and first digit ≠ 0) is 9 × P(9, n-1) for n ≤ 10.