Permutations and Combinations Hard Level

Pair as one unit → 5 units in a circle: 4! = 24

Pair can swap: × 2 = 48

A garland can be turned over: ÷ 2 = 24

Trap: 48 forgets that a garland can be flipped.

Answer: 24

Letters: A × 4, L × 2, H, B, D

All different: 5C3 = 10

Two alike + one other: A or L pair × 4 others = 8

Three alike: AAA = 1

Total = 19

Trap: 84 = 9C3 treats all 9 letters as different.

Answer: 19

Count ordered triples (a, b, c), 1–6, with a + b + c = 10

Patterns: 6+3+1, 6+2+2, 5+4+1, 5+3+2, 4+4+2, 4+3+3

Orders: 6 + 3 + 6 + 6 + 3 + 3 = 27

Answer: 27

Each divisor x of 72 fixes y

72 = 2³ × 3² → (3 + 1)(2 + 1) = 12 divisors

Trap: 6 counts unordered pairs.

Answer: 12

210 = 2 × 3 × 5 × 7

Each prime goes to one of a, b, c: 3⁴ = 81

Answer: 81

Couples as 3 units: 3! = 6

Each couple can swap: 2³ = 8

6 × 8 = 48

Trap: 6 or 24 forget the swaps.

Answer: 48

4 couple-units round a table: 3! = 6

Swaps: 2⁴ = 16

6 × 16 = 96

Trap: 384 treats the table as a row.

Answer: 96

Choose 3 of the 5 couples: 5C3 = 10

One partner from each: 2³ = 8

10 × 8 = 80

Trap: 120 = 10C3 allows couples.

Answer: 80

12C4 × 8C4 × 4C4 = 495 × 70 × 1 = 34,650

Groups unnamed: ÷ 3! = 5,775

Trap: 34,650 treats the groups as labelled.

Answer: 5,775

Alternate seatings: 3! × 4! = 144

Fix that man; the woman can sit in 2 neighbouring seats: 2 × 3! × 3! = 72

Not neighbours = 144 − 72 = 72

Answer: 72