Permutation & Combination

Before formulas, understand logic.

The AND Rule (Multiplication): If you must do Task A AND Task B, multiply their ways.
Example: 3 Shirts AND 2 Pants = $3 \times 2 = 6$ outfits.

The OR Rule (Addition): If you must do Task A OR Task B, add their ways.
Example: Go by Bus (3 types) OR Train (2 types) = $3 + 2 = 5$ ways.

The biggest confusion is 'Which formula to use?'. Apply the Order Test:

1. Change the order of selected items.
2. Does it make a new outcome?
- YES $\to$ Order Matters $\to$ Permutation (Arrangement)
- NO $\to$ Order Doesn't Matter $\to$ Combination (Selection)

Keywords:
- Permutation: Arrange, Line up, Code, Rank, Position.
- Combination: Select, Choose, Team, Group, Handshake.

Forget $nPr$! Use the Slot Method for arrangements.

Step 1: Draw empty slots (boxes) equal to the number of positions.
Step 2: Fill each slot with the number of options available for that position.
Step 3: Multiply them.

Visual Model:
Arranging 3 people on 5 chairs?
[5 options] $\times$ [4 options] $\times$ [3 options] = 60 ways.
(First person has 5 choices, next has 4, next has 3).

Use this when items must 'always keep together'.

The Trick:
1. Tie the 'together' items with a string and treat them as ONE unit.
2. Arrange the main units.
3. Multiply by the internal arrangement of the tied group.

Use this when items must 'never sit together'.

The Trick:
1. Arrange the 'others' first (creates gaps).
2. Place the specific items in the Gaps (spaces between adjacent items + ends).

Visual:
_ O _ O _ O _ (3 Others create 4 Gaps).
Select gaps using $nCr$, then arrange.

In a circle, there is no 'start' or 'end'.

Rule 1 (People/Distinct Items): $(n-1)!$
- Fix one person to create a reference point, then arrange remaining $(n-1)$.

Rule 2 (Necklace/Garland/Identical Flip): $\frac{(n-1)!}{2}$
- Clockwise and Anti-clockwise look same (flippable).

Find rank of word 'MAKE' in dictionary.
1. Write letters alphabetically: A, E, K, M.
2. Fix 'A' first $\to$ remaining 3! words. (Does target match? No, target starts with M).
3. Fix 'E' $\to$ 3! words.
4. Fix 'K' $\to$ 3! words.
5. Fix 'M' $\to$ Match! Now fix next letter 'A'...