Find the sum of the smallest and the greatest 3-digit numbers formed by using the digits 0, 1, 2, 3, 4 without any repetition of digits.
534
Finding Smallest and Greatest 3-Digit Numbers
The problem asks us to use the given digits 0, 1, 2, 3, and 4 to form the smallest possible 3-digit number and the greatest possible 3-digit number. A key constraint is that each digit can be used only once (without repetition).
Once we find both these numbers, we need to calculate their sum.
Forming the Smallest 3-Digit Number
To form the smallest 3-digit number using the digits 0, 1, 2, 3, 4 without repetition, we need to place the smallest possible digits in the higher value positions (hundreds, then tens, then units).
- The hundreds place cannot be 0 in a 3-digit number. So, the smallest non-zero digit available (1) must go in the hundreds place.
- After using 1, the remaining digits are 0, 2, 3, 4. To make the number smallest, the next smallest digit should go in the tens place. The smallest remaining digit is 0.
- After using 1 and 0, the remaining digits are 2, 3, 4. The smallest remaining digit (2) goes in the units place.
So, the smallest 3-digit number formed is 102.
Forming the Greatest 3-Digit Number
To form the greatest 3-digit number using the digits 0, 1, 2, 3, 4 without repetition, we need to place the largest possible digits in the higher value positions (hundreds, then tens, then units).
- The hundreds place should have the largest available digit, which is 4.
- After using 4, the remaining digits are 0, 1, 2, 3. To make the number greatest, the next largest digit should go in the tens place. The largest remaining digit is 3.
- After using 4 and 3, the remaining digits are 0, 1, 2. The largest remaining digit (2) goes in the units place.
So, the greatest 3-digit number formed is 432.
Calculating the Sum
Now we need to find the sum of the smallest 3-digit number (102) and the greatest 3-digit number (432).
Sum \( = \) Smallest number \( + \) Greatest number
Sum \( = 102 + 432 \)
Let's perform the addition:
\[ \begin{array}{@{}c@{\,}c@{}c} & 1 & 0 & 2 \\ + & 4 & 3 & 2 \\ \hline & 5 & 3 & 4 \\ \hline \end{array} \]The sum is 534.
Therefore, the sum of the smallest and the greatest 3-digit numbers formed by using the digits 0, 1, 2, 3, 4 without any repetition is 534.
| Description | Number |
|---|---|
| Digits used | 0, 1, 2, 3, 4 |
| Smallest 3-digit number | 102 |
| Greatest 3-digit number | 432 |
| Sum | \(102 + 432 = 534\) |
Revision Table: Key Concepts
| Concept | Explanation | How it applies here |
|---|---|---|
| Place Value | The value of a digit based on its position in a number (e.g., hundreds, tens, units). | Crucial for forming smallest/greatest numbers by placing digits in specific positions. |
| Smallest Number Formation | Place the smallest available non-zero digit in the highest place value, then the smallest remaining digits in descending order of place value. | Used to determine 102 from 0,1,2,3,4. |
| Greatest Number Formation | Place the largest available digit in the highest place value, then the largest remaining digits in descending order of place value. | Used to determine 432 from 0,1,2,3,4. |
| Without Repetition | Each digit from the given set can be used only once in the number being formed. | Ensures we don't use the same digit multiple times in 102 or 432. |
Additional Information: Number Formation with Digits
Understanding how to form the smallest and greatest numbers from a given set of digits is a fundamental concept in number systems. Here are some additional points:
- Role of Zero: When forming the smallest number, zero cannot be placed in the highest place value (like the hundreds place for a 3-digit number) as it would reduce the number of digits (e.g., 012 is 12, a 2-digit number). Zero is placed in the next highest place value to keep the number smallest.
- Role of Place Value: The digit in the leftmost position (highest place value) has the biggest impact on the number's magnitude. Placing the smallest possible non-zero digit there makes the number small, and placing the largest possible digit there makes the number large.
- Descending Order for Greatest: For the greatest number, after placing the largest digit in the highest place value, the remaining digits should be arranged in descending order from left to right (highest place value to lowest) to maximize the number's value.
- Ascending Order for Smallest (with care for zero): For the smallest number, after placing the smallest non-zero digit in the highest place value, the remaining digits should generally be arranged in ascending order from left to right. However, if zero is available, it usually goes in the second highest place value (tens place for a 3-digit number) to keep the number minimal while maintaining its number of digits.
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