Permutations and Combinations Hard Level

Seats with no two adjacent: (10 − 4 + 1)C4 = 7C4 = 35

Arrange the 4 people: 4! = 24

35 × 24 = 840

Trap: 35 forgets that the people are different.

Answer: 840

All routes: 8C4 = 70

Through X: 4C2 × 4C2 = 36

Avoiding X = 70 − 36 = 34

Trap: 36 is the number through X.

Answer: 34

All triangles: 10C3 = 120

Two sides on the decagon: 10

Exactly one side on it: 10 × 6 = 60

120 − 70 = 50

Trap: 70 is the number removed.

Answer: 50

Rectangles: 9C2 × 9C2 = 1,296

Squares: 204

1,296 − 204 = 1,092

Answer: 1,092

All: 10C5 = 252

Both on it: 8C3 = 56

252 − 56 = 196

Answer: 196

(2, 4): 10 × 5 = 50

(3, 3): 10 × 10 = 100

(4, 2): 5 × 10 = 50

Total = 200

Trap: 210 = 10C6 ignores the section rule.

Answer: 200

All 4-digit numbers: 9,000

With no 7: 8 × 9 × 9 × 9 = 5,832

9,000 − 5,832 = 3,168

Trap: 5,832 is the count without a 7.

Answer: 3,168

0 cannot appear (it would have to lead)

Choose 6 of 1–9; each choice gives one number: 9C6 = 84

Trap: 60,480 arranges them.

Answer: 84

The first three digits fix the number

First: 9 ways (not 0), second 10, third 10

9 × 10 × 10 = 900

Answer: 900

Boys first: 5! = 120

6 gaps around them; girls take 4 gaps in order: 6P4 = 360

120 × 360 = 43,200

Trap: 2,880 puts the girls in fixed gaps.

Answer: 43,200