Permutations and Combinations Exam Pattern Mix

A and B at the ends: 2 ways

Others in the middle: 3! = 6

2 × 6 = 12

Trap: 6 fixes A on the left.

Answer: 12

All 4-digit numbers: 9,000

All digits different: 9 × 9 × 8 × 7 = 4,536

9,000 − 4,536 = 4,464

Trap: 4,536 is the all-different count.

Answer: 4,464

President and VP: 10 × 9 = 90

Sub-committee from the other 8: 8C2 = 28

90 × 28 = 2,520

Answer: 2,520

1 is in, 2 is out; the other 8 elements are free

2⁸ = 256

Answer: 256

All subsets: 2⁶ = 64

Proper subsets leave out the set itself: 63

Answer: 63

Choose the 3 positions for the 1s: 8C3 = 56

Trap: 336 arranges identical 1s.

Answer: 56

Let f(n) be the count: f(1) = 2, f(2) = 3, f(n) = f(n − 1) + f(n − 2)

f(3) = 5, f(4) = 8, f(5) = 13, f(6) = 21

Answer: 21

26³ × 10³ = 17,576 × 1,000 = 1,75,76,000

Trap: 1,12,32,000 forbids repeats.

Answer: 1,75,76,000

Choose 4 of the 5 pairs: 5C4 = 5

One shoe from each: 2⁴ = 16

5 × 16 = 80

Trap: 130 is the number with at least one pair.

Answer: 80

All selections: 10C4 = 210

No pair: 80

210 − 80 = 130

Answer: 130