Understanding the Problem
To solve this, we need to form a 3-digit number using the set {1, 2, 3, 4, 5, 6} such that all digits are distinct and the final number is less than 500.
The Constraints
- Digit Pool: We have 6 available integers: {1, 2, 3, 4, 5, 6}.
- Distinctness: Each digit in our 3-digit number must be different (no repetition).
- Value Constraint: The number must be less than 500.
Step-by-Step Solution
Step 1: Analyze the Hundreds Place
For a 3-digit number to be less than 500, the hundreds place must be occupied by a digit smaller than 5. Looking at our set {1, 2, 3, 4, 5, 6}, the valid choices for the hundreds place are 1, 2, 3, or 4. That gives us 4 options for the hundreds place.
Step 2: Analyze the Tens Place
We have used one digit from our set of 6. Since the digits must be distinct, we have $6 - 1 = 5$ digits remaining. Therefore, there are 5 options for the tens place.
Step 3: Analyze the Units Place
We have now used two digits from the original set. This leaves us with $6 - 2 = 4$ digits remaining for the units place. Therefore, there are 4 options for the units place.
Step 4: Calculate the Total
Using the Fundamental Counting Principle, we multiply the number of independent choices for each position:
$$\text{Total} = (\text{Hundreds Options}) \times (\text{Tens Options}) \times (\text{Units Options})$$ $$\text{Total} = 4 \times 5 \times 4 = 80$$
There are 80 different 3-digit numbers that can be formed under these conditions.