Permutations and Combinations Medium Level

All = 7! = 5,040

Vowels A, I, E together: 5! × 3! = 720

Not all together = 5,040 − 720 = 4,320

Trap: 720 is the "together" count.

Answer: 4,320

Odd positions: 1, 3, 5 → place U and E: 3P2 = 6

Other 4 letters in the remaining 4 places: 4! = 24

6 × 24 = 144

Trap: 72 forgets that U and E can swap.

Answer: 144

3W 2M: 10 × 21 = 210

4W 1M: 5 × 7 = 35

5W: 1

Total = 246

Trap: 210 counts only exactly 3 women.

Answer: 246

0W: 7C5 = 21

1W: 7C4 × 5 = 175

2W: 7C3 × 5C2 = 350

Total = 546

Answer: 546

All groups = 10C4 = 210

Both included: 8C2 = 28

210 − 28 = 182

Trap: 28 is the count to remove.

Answer: 182

Pair as one unit → 6 units round a table: 5! = 120

Pair can swap: × 2 = 240

Answer: 240

All = 6! = 720

Together = 240

Not together = 480

Trap: 5,040 counts the circle like a row.

Answer: 480

First digit 4, 5, 6 or 7: 4 ways

Others: 4 × 3 × 2 = 24

4 × 24 = 96

Trap: 120 allows 3 to lead.

Answer: 96

Ending in 0: 5 × 4 × 3 = 60

Ending in 5: first digit not 0 → 4 × 4 × 3 = 48

Total = 108

Trap: 120 lets 0 lead in the second case.

Answer: 108

8 different letters, 5 of them vowels

First place: 5 ways; rest: 7! = 5,040

5 × 5,040 = 25,200

Answer: 25,200