Permutations and Combinations Medium Level

Aces: 4C2 = 6; other 3 from 48 non-aces: 48C3 = 17,296

6 × 17,296 = 1,03,776

Trap: 17,296 forgets to choose the aces.

Answer: 1,03,776

Fix one A; arrange A, R, R, N, G, E: 6!/2! = 360

Trap: 180 divides by 2 for the A that is already fixed.

Answer: 360

Letters: 26 × 25 = 650

Digits: 10 × 9 × 8 = 720

650 × 720 = 4,68,000

Trap: 6,76,000 allows repeats.

Answer: 4,68,000

Choose the other 3: 7C3 = 35

Arrange all 4: 4! = 24

35 × 24 = 840

Trap: 1,680 is with or without A.

Answer: 840

All: 7P4 = 840

Without M: 6P4 = 360

With M = 480

Answer: 480

5 rights and 3 ups in any order

8C3 = 56

Trap: 15 multiplies 5 × 3.

Answer: 56

A to P: 2 rights, 1 up → 3C1 = 3

P to B: 3 rights, 2 ups → 5C2 = 10

3 × 10 = 30

Trap: 13 adds the two parts.

Answer: 30

5 vertical lines, 4 horizontal lines

5C2 × 4C2 = 10 × 6 = 60

Trap: 12 counts only the small squares.

Answer: 60

1 × 1: 4 × 3 = 12

2 × 2: 3 × 2 = 6

3 × 3: 2 × 1 = 2

Total = 20

Trap: 18 misses the two large squares.

Answer: 20

Any 2 cards: 52C2 = 1,326

Same suit: 4 × 13C2 = 312

1,326 − 312 = 1,014

Trap: 2,028 counts each pair twice.

Answer: 1,014