How To Find The Number Of 3-Digit Numbers With Unique Digits From 0 To 5

This is a classic problem in combinatorics, a branch of mathematics dealing with counting, arrangement, and combination of objects. Let's break down how to solve this problem step-by-step.

Understanding the Problem

Our main keyword is three-digit numbers with unique digits. We have six digits available: 0, 1, 2, 3, 4, and 5. The challenge is to figure out how many different three-digit numbers we can create using these digits, with the crucial condition that no digit can be repeated within a single number. For instance, 123 is valid, but 121 is not because the digit '1' is repeated.

Breaking Down the Constraints

The constraints of the problem significantly impact our approach. The most important constraint is that the numbers must be three-digit numbers. This means the first digit (the hundreds place) cannot be zero. If we place zero in the hundreds place, the number effectively becomes a two-digit number. This constraint makes the problem slightly more complex than a simple permutation.

The second key constraint is that the digits must be unique. This means that once a digit is used in a place value (hundreds, tens, or ones), it cannot be used again. This eliminates possibilities like 111, 223, or 454. This restriction fundamentally changes the calculation because the number of available digits decreases as we fill each place value.

Solving the Problem: A Step-by-Step Approach

To solve this problem, we'll use the fundamental counting principle, which states that if there are 'm' ways to do one thing and 'n' ways to do another, then there are m * n ways to do both. We'll apply this principle to each digit place in our three-digit number.

1. Choosing the Hundreds Digit

Let's start with the hundreds digit. We have six digits (0, 1, 2, 3, 4, and 5) to choose from, but we cannot use 0. Therefore, we have 5 options for the hundreds digit (1, 2, 3, 4, or 5). This is a crucial first step as it considers the three-digit number requirement.

2. Choosing the Tens Digit

Now, let's move to the tens digit. We've already used one digit for the hundreds place, leaving us with five remaining digits. Importantly, we can now use 0 for the tens digit. So, we have 5 options for the tens digit. This is because we initially had six digits, used one for the hundreds place, and now we can include 0 in our choices.

3. Choosing the Units Digit

Finally, let's consider the units digit. We've used two digits already (one for the hundreds place and one for the tens place), leaving us with four remaining digits. Therefore, we have 4 options for the units digit. Each digit we select reduces the available options for subsequent digits due to the unique digit constraint.

4. Applying the Fundamental Counting Principle

To find the total number of three-digit numbers, we multiply the number of options for each digit place:

Total numbers = (Options for hundreds digit) * (Options for tens digit) * (Options for units digit) Total numbers = 5 * 5 * 4 = 100

Therefore, there are 100 three-digit numbers with unique digits that can be formed from the digits 0, 1, 2, 3, 4, and 5.

Why Other Options Are Incorrect

It's helpful to understand why the other answer choices are incorrect. This reinforces the correct methodology and highlights common mistakes.

• A) 125: This number is likely a result of not properly accounting for the restriction on the hundreds digit (not being able to use 0) or making a mistake in the multiplication.
• B) 216: This number might arise from considering all possible permutations of three digits from the six available (6 * 6 * 6), but this doesn't account for the non-repetition of digits and the restriction on the hundreds digit.
• D) 180: This number could result from an error in applying the counting principle or misinterpreting one of the constraints.

Key Concepts Revisited

Let's reinforce the key concepts used in solving this problem:

• Combinatorics: The branch of mathematics dealing with counting, arrangements, and combinations.
• Fundamental Counting Principle: If there are 'm' ways to do one thing and 'n' ways to do another, then there are m * n ways to do both.
• Permutations: Arrangements of objects in a specific order (though this problem involves a slight variation due to the constraints).
• Constraints: Limitations or conditions that affect the solution (in this case, the three-digit requirement and the unique digit requirement).

Similar Problems and Extensions

This problem is a foundation for more complex combinatorics questions. Here are a few ways the problem could be extended:

• Four-digit numbers: How many four-digit numbers with unique digits can be formed using the same digits?
• Different digit sets: What if we used a different set of digits, like 0, 2, 4, 6, and 8?
• Odd/Even numbers: How many three-digit odd numbers with unique digits can be formed?
• Numbers within a range: How many three-digit numbers between 200 and 400 with unique digits can be formed?

Solving these variations requires a similar logical approach but might introduce additional constraints or considerations. The key is to break the problem down into smaller steps, consider the constraints carefully, and apply the fundamental counting principle.

Conclusion

The answer to the question "How many three-digit numbers with unique digits can be formed using the digits 0, 1, 2, 3, 4, and 5?" is C) 100. This problem showcases the importance of understanding the fundamental counting principle and carefully considering constraints when solving combinatorial problems. By breaking down the problem into smaller steps and systematically analyzing each digit place, we can arrive at the correct solution. Remember, the core of solving these problems lies in understanding the restrictions and applying the right counting techniques. Always double-check your work and consider why other options might be incorrect to solidify your understanding.