How Many Three-Digit Numbers Can Be Formed Using Digits 4, 7, and 0?

Forming three-digit numbers with the digits 4, 7, and 0 can be approached in multiple ways, considering whether digits can be repeated and whether to include negative numbers. Let's explore the different permutations and combinations.

Without Repeating Digits

The first challenge is to understand how we can arrange the digits 4, 7, and 0 without repeating any digit in a three-digit number. The hundreds place cannot be 0, as it would then be a two-digit number. This constraint significantly reduces the possible combinations:

If 4 is in the hundreds place, the valid numbers are 407 and 470. If 7 is in the hundreds place, the valid numbers are 704 and 740.

Therefore, there are a total of 4 possible three-digit numbers without repeating digits:

407, 470, 704, 740

With Digits Repeated

When digits can be repeated, the possibilities expand greatly:

The hundreds place can be either 4 or 7 (2 options). The tens and units places each have 3 options (0, 4, 7).

So, the total number of permutations, considering duplicates, is:

2 (for the hundreds place) times; 3 (for the tens place) times; 3 (for the units place) 18

The complete list of numbers includes:

400, 404, 440, 444, 447, 470, 474, 477 700, 704, 740, 744, 747, 770, 774, 777

Consideration of Negative Numbers

The above solutions consider only positive numbers. However, negative numbers can also have three digits. For each positive number, there is a corresponding negative number. This doubles the count:

Positive numbers: 407, 470, 704, 740 Negative numbers: -407, -470, -704, -740

When digits can be repeated, the total number of possible negative numbers is also 18, bringing the total count to:

4 (positive) 4 (negative) 8 (without repetition) 20 (positive) 18 (negative) 36 (with repetition)

Conclusion and Further Exploration

This analysis reveals that the number of three-digit numbers that can be formed using the digits 4, 7, and 0, including negative numbers, depends on whether digits can be repeated. Understanding permutations and combinations is crucial in this context to ensure a comprehensive solution:

Without repetition: 4 numbers (positive) 4 numbers (negative) 8 total With repetition: 20 numbers (positive) 18 numbers (negative) 36 total

By exploring such problems, one can gain insight into the intricacies of combinatorial mathematics and the importance of considering all possible variations.