To determine how many three-digit numbers less than 500 can be formed from the digits 1, 2, 3, 4, 5, and 6, where each digit must be different, follow these steps:

Step-by-Step Solution:

1. Identify the Range:

   • A three-digit number less than 500 has its hundreds digit restricted to 1, 2, 3, or 4.
   • Therefore, the hundreds place can be any of 1, 2, 3, or 4.

2. Determine the Number of Combinations for Each Hundreds Digit:

   • Hundreds place = 1:

     • Tens place can be 2, 3, 4, 5, or 6 (5 options left).
     • Units place can be any of the remaining 4 digits.
     • Total combinations = $5 \times 4 = 20$.

   • Hundreds place = 2:

     • Tens place can be 1, 3, 4, 5, or 6 (5 options left).
     • Units place can be any of the remaining 4 digits.
     • Total combinations = $5 \times 4 = 20$.

   • Hundreds place = 3:

     • Tens place can be 1, 2, 4, 5, or 6 (5 options left).
     • Units place can be any of the remaining 4 digits.
     • Total combinations = $5 \times 4 = 20$.

   • Hundreds place = 4:

     • Tens place can be 1, 2, 3, 5, or 6 (5 options left).
     • Units place can be any of the remaining 4 digits.
     • Total combinations = $5 \times 4 = 20$.

3. Add Up All Combinations:

   • Total combinations for all four cases: $20 + 20 + 20 + 20 = 80$

Conclusion:

There are $80$ different three-digit numbers less than 500 that can be formed from the digits 1, 2, 3, 4, 5, and 6, with no repeated digits.

Would you like more details or have any questions?

Here are 5 additional questions you might consider:

1. How many four-digit numbers can be formed using the digits 1, 2, 3, and 4 without repetition?
2. What is the number of permutations of the digits 1, 2, 3, 4, and 5 taken three at a time?
3. How many two-digit numbers can be formed from the digits 5, 6, 7, 8, and 9?
4. What is the number of combinations of the digits 1, 2, 3, 4, and 5 taken two at a time?
5. How many three-digit numbers can be formed using only the digits 1, 2, and 3 without repetition?

Tip: When calculating combinations and permutations, remember to consider whether the order of selection matters (permutations) or not (combinations).