How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?
336 telephone numbers can be constructed.
Explanation
This question tests the application of the multiplication principle with constraints. The key insight is that since the first two digits are fixed (6 and 7), we only need to determine how many ways we can fill the remaining three positions. The constraint 'no digit appears more than once' means we cannot reuse 6 or 7, leaving us with 8 available digits for the remaining positions.
Solution Steps
Step 1: The telephone number has 5 digits. The first two digits are fixed as 6 and 7 respectively.
Step 2: We need to fill the remaining 3 positions (3rd, 4th, and 5th digits) using digits from 0 to 9.
Step 3: Since digits 6 and 7 are already used and no digit can be repeated, we have 10 - 2 = 8 digits available for the remaining positions.
Step 4: For the 3rd position: 8 choices available (any of the remaining digits).
Step 5: For the 4th position: 7 choices available (one digit used in 3rd position).
Step 6: For the 5th position: 6 choices available (two digits already used).
Step 7: By the multiplication principle, total numbers = × 6 = 336.