Core Laws & Golden Rules

Law 1: Multiplication & Addition Principle

• Multiplication Rule (AND): If task A can be done in m ways and task B in n ways, total = m × n

• Addition Rule (OR): If either task A (m ways) or task B (n ways), total = m + n

Example: You have 3 shirts and 2 pants. How many outfits? Solution: 3 × 2 = 6 outfits

CAT Shortcut/Trap:

• Many students mix AND/OR in questions. so always check if events are independent (AND) or mutually exclusive (OR).

Law 2: Permutation vs Combination

• Permutation (P): Arrangement matters → nPr = n!/(n−r)!

• Combination (C): Selection order doesn’t matter → nCr = n!/r!(n−r)!

Example: 5 digits {1,2,3,4,5}, make 3-digit numbers: Permutation: 5P3 = 5 × 4 × 3 = 60

CAT Shortcut/Trap:

• Questions may look like combination but require arranging after selection. Always ask: Does order matter?

Law 3: Repetition & Restrictions

• With repetition: Use n^r

• With restrictions: Total − Bad = Good

Example: Arrange letters of “BANANA” so no two N’s together. Solution: Total ways = 6! / (3! × 2!) = 60 → Place N’s in gaps between A’s (Gap Method)

CAT Shortcut:

• Gap Method: Fix one group, then insert restricted elements in the remaining positions. This avoids overcounting.

Law 4: Circular Arrangements

• n people around a circle = (n−1)!

• If clockwise & anticlockwise are identical = (n−1)! / 2

Example: 6 boys sit around a table. Two must sit together? Treat pair as one unit → 5! × 2 = 240 ways

CAT Shortcut:

• Always check if direction matters (clockwise vs anticlockwise). Reduces calculation by half if symmetrical.

Tricks at one go:

1. Total − Bad: Great for forbidden arrangements (e.g., vowels together).

2. Gap Method: Space out restricted elements.

3. Factorial Cancellation: Simplify permutations before multiplying.

4. Symmetry in Combinations: nCr = nC(n−r)→ reduces calculation.