Permutations and Combinations Medium Level

nC2 = 66 → n(n − 1) = 132 = 12 × 11

n = 12

Answer: 12

Girls as one block + 5 boys → 6! = 720

Girls inside the block: 3! = 6

720 × 6 = 4,320

Answer: 4,320

Patterns: BGBG… or GBGB… → 2

Boys 4! × girls 4! = 576 per pattern

2 × 576 = 1,152

Trap: 576 uses only one pattern.

Answer: 1,152

Seat the boys in a circle: 3! = 6

Girls in the 4 gaps: 4! = 24

6 × 24 = 144

Trap: 1,152 treats the table like a row.

Answer: 144

Apples: 0–3 (4 ways), bananas 5 ways, mangoes 6 ways

4 × 5 × 6 = 120, minus choosing nothing

= 119

Trap: 120 includes the empty choice.

Answer: 119

360 = 2³ × 3² × 5

(3 + 1)(2 + 1)(1 + 1) = 24

Answer: 24

Each book is in or out: 2¹⁰ = 1,024

Remove choosing none: 1,023

Trap: 1,024 includes the empty choice.

Answer: 1,023

Order of subjects: 2!

Maths among themselves 3!, physics 2!

2 × 6 × 2 = 24

Trap: 12 forgets that the subjects can swap.

Answer: 24

Choose 4 for the first table: 8C4 = 70

Each table: 3! circular arrangements

70 × 6 × 6 = 2,520

Trap: 1,260 treats the tables as identical.

Answer: 2,520

10C5 = 252 picks one team; the other is fixed

Each split is counted twice → 126

Trap: 252 counts each split twice.

Answer: 126