Permutations and Combinations Exam Pattern Mix

10 letters: M, A, N, E each twice

10! ÷ (2!)⁴ = 36,28,800 ÷ 16 = 2,26,800

Trap: 4,53,600 divides for only three pairs.

Answer: 2,26,800

7 letters: S × 3, C × 2

7! ÷ (3! × 2!) = 420

Trap: 840 divides only for the Ss.

Answer: 420

SSS as one unit + U, C, C, E → 5 units with C twice

5!/2! = 60

Trap: 120 forgets the repeated C.

Answer: 60

All: 10C4 = 210

No doctor: 6C4 = 15

210 − 15 = 195

Answer: 195

4M 2W: 35 × 6 = 210

5M 1W: 21 × 4 = 84

6M: 7

Total = 301

Trap: 210 takes only 4 men.

Answer: 301

Last digit odd: 5 ways

First digit: not 0 and not the last → 8

Middle: 8 left

5 × 8 × 8 = 320

Trap: 360 lets 0 lead.

Answer: 320

a + b + c = 5 with a ≥ 1 → a′ + b + c = 4

(4 + 2)C2 = 15

Trap: 21 lets the first digit be 0.

Answer: 15

1 flag: 6; 2 flags: 6 × 5 = 30; 3 flags: 6 × 5 × 4 = 120

Total = 156

Trap: 41 counts selections, not orders.

Answer: 156

First digit fixed as 9

Remaining 6 places from the other 9 digits: 9P6 = 60,480

Trap: 5,31,441 = 9⁶ allows repeats.

Answer: 60,480

4 choices for each: 4⁸ = 65,536

Trap: 32 adds 8 × 4.

Answer: 65,536