Applying Permutations in Context

1) How many 4-letter arrangements can be made from the letters in the word "MATH" without repetition?

8

12

24

16

There are 4P4 = 4! = 24 possible arrangements.

Explanation

There are 4P4 = 4! = 24 possible arrangements.

2) How many ways can the top 3 winners be chosen from 10 contestants if order matters?

720

120

210

60

10P3 = 10 × 9 × 8 = 720.

Explanation

10P3 = 10 × 9 × 8 = 720.

3) How many 3-digit numbers can be formed using the digits 1, 2, 3, 4, 5 if no digit repeats?

30

125

20

60

5P3 = 5 × 4 × 3 = 60.

Explanation

5P3 = 5 × 4 × 3 = 60.

4) A password consists of 5 unique letters chosen from 8. How many different passwords can be formed?

1680

6720

3360

5040

8P5 = 8 × 7 × 6 × 5 × 4 = 6720.

Explanation

8P5 = 8 × 7 × 6 × 5 × 4 = 6720.

5) How many different podium orders are possible for a 6-person race (gold, silver, bronze)?

90

120

60

36

6P3 = 6 × 5 × 4 = 120.

Explanation

6P3 = 6 × 5 × 4 = 120.

6) How many 2-letter “initials” can be made from the alphabet if letters cannot repeat?

702

676

600

650

26P2 = 26 × 25 = 650.

Explanation

26P2 = 26 × 25 = 650.

7) How many permutations are there when choosing 4 books from a shelf of 7 to line up on a table?

420

840

210

5040

7P4 = 7 × 6 × 5 × 4 = 840.

Explanation

7P4 = 7 × 6 × 5 × 4 = 840.

8) A code uses 2 distinct letters followed by 2 distinct digits (0–9). How many codes are possible?

25,000

52,000

40,000

58,500

26P2 = 26 × 25 = 650; 10P2 = 10 × 9 = 90; total 650 × 90 = 58,500.

Explanation

26P2 = 26 × 25 = 650; 10P2 = 10 × 9 = 90; total 650 × 90 = 58,500.

9) How many arrangements of 6 people are possible if they line up for a photo?

360

720

600

840

6P6 = 6! = 720.

Explanation

6P6 = 6! = 720.

10) How many ways can a president, vice president, and secretary be chosen from 8 people if no one holds more than one office?

336

210

56

168

8P3 = 8 × 7 × 6 = 336.

Explanation

8P3 = 8 × 7 × 6 = 336.