How many 4-digit numbers are there with no digit repeated?
The total number of 4-digit numbers with no digit repeated is 4536.
Explanation
This question applies the multiplication principle from the chapter. A 4-digit number ranges from 1000 to 9999. The key constraint is that the first digit cannot be 0 (otherwise it becomes a 3-digit number). Using the multiplication principle as demonstrated in Example 2 (flag signals) and Exercise 6.1 Question 1, we multiply the number of choices at each position. The calculation follows: thousands place has 9 options (1-9), hundreds place has 9 remaining options (0-9 excluding the first digit), tens place has 8 options, and units place has 7 options.
Solution Steps
Step 1: For a 4-digit number, the thousands place cannot be 0. It can be filled by any digit from 1 to 9. Number of ways = 9.
Step 2: The hundreds place can be filled by any digit from 0 to 9 except the digit already used in thousands place. Number of ways = 9.
Step 3: The tens place can be filled by any remaining digit from 0 to 9 except the two digits already used. Number of ways = 8.
Step 4: The units place can be filled by any remaining digit from 0 to 9 except the three digits already used. Number of ways = 7.
Step 5: By the multiplication principle, total number of 4-digit numbers = × = 4536.