Click on a question to see the answer and explanation.
▶1. In how many ways can 4 different books be arranged on a shelf?
Answer: 24
Explanation:
- Step 1: The number of ways to arrange n different items is n!
- Step 2: 4! = 4 × 3 × 2 × 1 = 24
▶2. How many ways can 3 students be seated in a row of 5 chairs?
Answer: 60
Explanation:
- Step 1: Number of ways = 5P3 = 5! / (5-3)!
- Step 2: 5! = 120, 2! = 2
- Step 3: 120 / 2 = 60
▶3. How many 3-digit numbers can be formed using the digits 1, 2, 3, 4 if no digit is repeated?
Answer: 24
Explanation:
- Step 1: Number of ways = 4P3 = 4! / (4-3)! = 24
▶4. In how many ways can the letters of the word "MATH" be arranged?
Answer: 24
Explanation:
- Step 1: All letters are different, so 4! = 24
▶5. How many ways can 5 people sit around a circular table?
Answer: 24
Explanation:
- Step 1: For a circle, (n-1)! ways
- Step 2: (5-1)! = 4! = 24
▶6. How many 4-letter words can be formed from the letters of "MOON"?
Answer: 12
Explanation:
- Step 1: O repeats twice. Number of arrangements = 4! / 2! = 12
▶7. In how many ways can 3 boys and 2 girls be arranged in a row?
Answer: 120
Explanation:
- Step 1: Total people = 5
- Step 2: 5! = 120
▶8. How many ways can the letters of the word "LEVEL" be arranged?
Answer: 30
Explanation:
- Step 1: L and E repeat twice each.
- Step 2: Number of arrangements = 5! / (2! × 2!) = 120 / 4 = 30
▶9. How many ways can 4 people be seated in a row if two specific people must sit together?
Answer: 12
Explanation:
- Step 1: Treat the two as a block: now 3 blocks to arrange = 3! = 6
- Step 2: The two can switch places: 2! = 2
- Step 3: Total = 6 × 2 = 12
▶10. In how many ways can the letters of "BANANA" be arranged?
Answer: 60
Explanation:
- Step 1: A repeats 3 times, N repeats 2 times.
- Step 2: Number of arrangements = 6! / (3! × 2!) = 720 / 12 = 60
▶11. How many 3-digit numbers can be formed from 1, 2, 3, 4, 5 if repetition is allowed?
Answer: 125
Explanation:
- Step 1: Each digit can be chosen in 5 ways.
- Step 2: 5 × 5 × 5 = 125
▶12. How many ways can 6 people be arranged in a row if two specific people must not sit together?
Answer: 480
Explanation:
- Step 1: Total arrangements = 6! = 720
- Step 2: Arrangements with the two together: treat as a block, so 5! × 2! = 240
- Step 3: Not together = 720 - 240 = 480
▶13. In how many ways can 5 different colored beads be arranged in a necklace?
Answer: 12
Explanation:
- Step 1: For a necklace, arrangements = (n-1)! / 2
- Step 2: (5-1)! / 2 = 24 / 2 = 12
▶14. How many ways can 4 boys and 3 girls be arranged in a row so that no two girls are together?
Answer: 1440
Explanation:
- Step 1: Arrange boys: 4! = 24
- Step 2: Gaps between boys = 5, choose 3 for girls: 5P3 = 60
- Step 3: Total = 24 × 60 = 1440
▶15. How many ways can 7 people be seated at a round table if two must not sit together?
Answer: 3600
Explanation:
- Step 1: Total circular arrangements = (7-1)! = 720
- Step 2: Arrangements with two together: treat as a block, so (6-1)! × 2! = 120 × 2 = 240
- Step 3: Not together = 720 - 120 = 600 (for a round table, but if it's a row, use 3600 for 7! - 6! × 2!)
▶16. How many ways can the letters of "SUCCESS" be arranged?
Answer: 420
Explanation:
- Step 1: S repeats 3 times, C repeats 2 times.
- Step 2: Number of arrangements = 7! / (3! × 2!) = 5040 / 12 = 420
▶17. In how many ways can 3 boys and 2 girls be arranged in a row so that the girls are always together?
Answer: 48
Explanation:
- Step 1: Treat the girls as a block: now 4 blocks to arrange = 4! = 24
- Step 2: Girls can switch places: 2! = 2
- Step 3: Total = 24 × 2 = 48
▶18. How many ways can 6 books be arranged on a shelf if 2 specific books must be at the ends?
Answer: 48
Explanation:
- Step 1: 2 books at ends: 2! ways
- Step 2: Arrange remaining 4 books: 4! = 24
- Step 3: Total = 2 × 24 = 48
▶19. How many ways can 5 people be arranged in a row if one person must always be at the end?
Answer: 24
Explanation:
- Step 1: Fix one person at the end: 1 way
- Step 2: Arrange remaining 4: 4! = 24
▶20. In how many ways can the letters of "APPLE" be arranged?
Answer: 60
Explanation:
- Step 1: P repeats twice.
- Step 2: Number of arrangements = 5! / 2! = 120 / 2 = 60