Permutations and Combinations Exam Pattern Mix

By symmetry A is left of B in half the orders

7! ÷ 2 = 2,520

Trap: 720 makes them adjacent.

Answer: 2,520

A, B, C can be in 3! orders, equally often

6! ÷ 3! = 120

Answer: 120

Choose any 4 of the 10 digits; arranged in decreasing order they give one number

0 can only come last, which is fine: 10C4 = 210

Trap: 126 = 9C4 wrongly leaves out 0.

Answer: 210

Last two digits must form a multiple of 4: 12, 24, 32, 44, 52 → 5

First digit: 5 ways

5 × 5 = 25

Answer: 25

Each digit appears 2! = 2 times in each place

2 × (1 + 3 + 5) × 111 = 1,998

Answer: 1,998

Choose the suit: 4; choose 5 of its 13 cards: 13C5 = 1,287

4 × 1,287 = 5,148

Trap: 1,287 forgets the 4 suits.

Answer: 5,148

Rank for the three: 13, cards 4C3 = 4

Rank for the pair: 12, cards 4C2 = 6

13 × 4 × 12 × 6 = 3,744

Trap: 1,872 treats the two ranks as unordered.

Answer: 3,744

7P5 = 7 × 6 × 5 × 4 × 3 = 2,520

Trap: 21 only chooses the seats.

Answer: 2,520

Driver: 2 ways

Other 4 seats: 4! = 24

2 × 24 = 48

Trap: 120 lets anyone drive.

Answer: 48

Settings: 10³ = 1,000, of which one opens the lock

Unsuccessful = 999

Trap: 1,000 includes the right setting.

Answer: 999