Q5. How many 3-digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5 if no digit is repeated?
The hundred’s place cannot be 0: 5 ways. The ten’s place then has 5 ways (0 is now allowed) and the unit’s place 4: 5 × 5 × 4 = 100.
Q6. A restaurant offers 4 starters, 5 main courses and 3 desserts. In how many ways can a three-course meal (one of each) be chosen? In how many if a diner may skip the dessert?
4 × 5 × 3 = 60. If the dessert may be skipped, the dessert stage has 4 options (3 desserts or none): 4 × 5 × 4 = 80.
Q7. How many 4-digit numbers have all four digits odd?
Each of the 4 places has 5 choices (1, 3, 5, 7, 9): 54 = 625.
Q8. How many 3-digit numbers have at least one repeated digit?
All 3-digit numbers: 9 × 10 × 10 = 900. With no repeated digit: 9 × 9 × 8 = 648. So 900 − 648 = 252 have a repeated digit.