Permutations and Combinations Medium Level

9 letters: T three times, I twice

9! ÷ (3! × 2!) = 3,62,880 ÷ 12 = 30,240

Trap: 60,480 divides only for the Ts.

Answer: 30,240

Choose from the other 9: 9C3 = 84

Answer: 84

Stars and bars with each ≥ 1: (8 − 1)C(3 − 1) = 7C2 = 21

Trap: 45 = 10C2 allows a child to get none.

Answer: 21

Stars and bars: (10 + 2)C2 = 12C2 = 66

Trap: 36 = 9C2 requires each to be at least 1.

Answer: 66

n(n − 3)/2 = 44 → n(n − 3) = 88 = 11 × 8

n = 11

Answer: 11

All triples = 12C3 = 220

Collinear triples (no triangle) = 5C3 = 10

220 − 10 = 210

Trap: 220 forgets the collinear points.

Answer: 210

All pairs = 66

The 5 collinear points give 5C2 = 10 pairs but only 1 line

66 − 10 + 1 = 57

Trap: 56 forgets to add back that one line.

Answer: 57

Choose 2 lines from each set: 4C2 × 5C2 = 6 × 10 = 60

Trap: 20 multiplies 4 × 5.

Answer: 60

9 lines each way; choose 2 of each

9C2 × 9C2 = 36 × 36 = 1,296

Trap: 204 counts only squares.

Answer: 1,296

1² + 2² + … + 8² = 204

Trap: 64 counts only the small squares.

Answer: 204