Permutations and Combinations Hard Level

Vowels E, E, E, I, I as one block; consonants N, N, N, G, G, R

Units: block + 6 → 7!/(3! 2!) = 420

Inside the block: 5!/(3! 2!) = 10

420 × 10 = 4,200

Trap: 420 forgets to arrange the vowels.

Answer: 4,200

11 letters: E 3, N 3, G 2, I 2, R 1

11! ÷ (3! 3! 2! 2!) = 2,77,200

Trap: 5,54,400 divides for one I/G pair only.

Answer: 2,77,200

Letters in order: E H M O R T

Starting with E or H: 2 × 5! = 240

M then E or H: 2 × 4! = 48

MO then E, H or R: 3 × 3! = 18

MOT then E: 1 × 2! = 2

MOTHER itself: + 1

Rank = 240 + 48 + 18 + 2 + 1 = 309

Trap: 308 forgets to count the word itself.

Answer: 309

Each digit appears 3! = 6 times in each place

Sum = 6 × (1 + 2 + 3 + 4) × 1,111 = 66,660

Answer: 66,660

Derangements of 5: 5! (1 − 1 + 1/2 − 1/6 + 1/24 − 1/120)

= 60 − 20 + 5 − 1 = 44

Answer: 44

All ways: 3⁵ = 243

Remove one box empty: 3 × 2⁵ = 96, add back two empty: 3

243 − 96 + 3 = 150

Trap: 243 allows empty boxes.

Answer: 150

Put 2 in each first; 2 balls remain

Share 2 among 4 boxes freely: (2 + 3)C3 = 10

Trap: 84 = 9C3 needs only one ball in each.

Answer: 10

Let x = 2 + x′: x′ + y + z + w = 18

(18 + 3)C3 = 21C3 = 1,330

Trap: 1,771 = 23C3 ignores x ≥ 2.

Answer: 1,330

Choosing r non-consecutive from n: (n − r + 1)Cr

(10 − 4 + 1)C4 = 7C4 = 35

Answer: 35

All = 7! = 5,040

Neighbours: 6! × 2 = 1,440

5,040 − 1,440 = 3,600

Answer: 3,600