Counting questions look simple but hide subtle traps. The key skill is not memorizing formulas — it's correctly deciding which formula to use, or whether you need a formula at all.

A) Start Here: The One Decision That Matters

Before touching a formula, ask: does order matter?

• Choosing a president, VP, and secretary from 10 people → order matters (different roles) → Permutation
• Choosing a 3-person committee from 10 people → order doesn't matter (same role) → Combination
• Arranging books on a shelf → order matters (position matters) → Permutation
• Selecting flavors for a sampler box → order doesn't matter → Combination

B) Fundamental Counting Principle

If task 1 can be done in m ways and task 2 in n ways (independently), both together can be done in m × n ways.

Example: 3 shirt colors and 4 pant styles → 3 × 4 = 12 possible outfits.

Example: A PIN uses 4 digits (each 0–9), no restriction → 10 × 10 × 10 × 10 = 10,000 possible PINs.

Example: Same PIN, no digit repeated → 10 × 9 × 8 × 7 = 5,040 PINs (each slot reduces options by 1).

C) Factorials — The Building Block

n! (n factorial) = n × (n−1) × (n−2) × … × 2 × 1. It counts the ways to arrange n distinct items.

• 0! = 1 (by definition)
• 1! = 1  |  2! = 2  |  3! = 6  |  4! = 24  |  5! = 120  |  6! = 720  |  7! = 5,040

Key identities: n! = (n − 1)! × n   and   (n + 1)! = n! × (n + 1). These let you simplify factorial fractions without computing large values.

Example: 8! / 7! = 7! × 8 / 7! = 8.   10! / 8! = 10 × 9 = 90.

Example: 5 different books on a shelf → 5! = 120 arrangements.

D) Permutations (Order Matters)

P(n, r) = n! / (n − r)! — choosing r items from n when order matters.

Intuition: you fill r slots one at a time. First slot has n choices, second has n−1, …, r-th slot has n−r+1. Multiply them together.

• Example: Gold, silver, and bronze medals for 8 runners → P(8,3) = 8×7×6 = 336 ways.
• Example: 4-letter code from 26 letters, no repeats → P(26,4) = 26×25×24×23 = 358,800 codes.

Arrangements with identical items: n! / (k₁! × k₂! × …) for each group of identical items.

• Example: Arrange the letters in MISSISSIPPI (11 letters: 4 S, 4 I, 2 P, 1 M) → 11! / (4!×4!×2!×1!) = 34,650 arrangements.
• Example: Arrange AABB → 4!/(2!×2!) = 24/4 = 6 arrangements: AABB, ABAB, ABBA, BABA, BBAA, BAAB.

E) Combinations (Order Doesn't Matter)

C(n, r) = n! / (r! × (n − r)!) — choosing r items from n when order is irrelevant.

Intuition: start with P(n,r) but divide by r! to remove duplicate orderings of the same group.

• Example: 3-person committee from 8 people → C(8,3) = 8!/(3!×5!) = (8×7×6)/(3×2×1) = 336/6 = 56 committees.
• Example: Choose 2 toppings from 5 options → C(5,2) = (5×4)/(2×1) = 10 combinations.
• Symmetry: C(n,r) = C(n, n−r). Choosing 3 from 8 = choosing which 5 to leave out → same count.
• Special values: C(n,0) = 1  |  C(n,1) = n  |  C(n,2) = n(n−1)/2.

F) Special Cases

• Circular arrangements: (n−1)! ways. Fix one person's position to remove rotational symmetry, then arrange the rest.
  Example: 5 people seated at a round table → 4! = 24 ways.
• Items that must stay together: glue them into one unit. Arrange the units (treat as m units), then arrange within the glued unit.
  Example: 5 people in a row, A and B must be adjacent → treat AB as 1 unit: 4 units total = 4! = 24 arrangements × 2 (AB or BA) = 48 ways.
• Items that cannot be together: total − arrangements where they ARE together.
  Example: 5 people in a row, A and B not adjacent → 5! − 48 = 120 − 48 = 72 ways.
• Selections by category: multiply the combinations for each independent category.
  Example: Choose 2 men from 4 AND 3 women from 5 → C(4,2) × C(5,3) = 6 × 10 = 60 ways.
• At least one of type X: total combinations − combinations with zero X.
  Example: Choose 3 from 4 red and 5 blue, at least 1 red → C(9,3) − C(5,3) = 84 − 10 = 74 ways.

G) Step-by-Step Approach for Any Counting Problem

1. Read carefully — identify what you are counting (arrangements, selections, codes?).
2. Ask: does order matter? → Permutation or Combination.
3. Check for restrictions (must be together, cannot be together, at least one, identical items).
4. Break into independent sub-tasks and multiply their counts.
5. Use complement (total − unwanted) when "at least" or "at most" conditions appear.