6. Using each of the digits $1,2,3,4,5$ and 6 exactly once, Ivan needs to write two numbers whose sum is 750. How many different possibilities are there? Determine the possibility for which the difference between the two written numbers is the largest possible.
6. Using each of the digits 1, 2, 3, 4, 5, and 6 exactly once, Ivan needs to write two numbers whose sum is 750. How many different possibilities are there? Determine the possibility for which the difference between the two written numbers is the largest possible.

## Solution.

If one of the numbers were two-digit (or one-digit), the other would be four-digit (or five-digit), making the sum greater than 750. Therefore, both numbers must be three-digit.

1 POINT

Their sum will end in zero only if the digits in the units place are 4 and 6.

From the remaining digits, the sum of 7 in the hundreds place can only be achieved with the digits 5 and 2.

2 POINTS

Note that for the tens place, the remaining digits are 1 and 3. Since the sum of the units digits is 10, the sum of the tens digits is indeed 5. The desired pairs of numbers are:

| Larger number | 536 | 534 | 516 | 514 |
| :--- | :--- | :--- | :--- | :--- |
| Smaller number | 214 | 216 | 234 | 236 |

There are a total of four possibilities.

2 POINTS

The largest difference is achieved when the minuend is the largest possible and the subtrahend is the smallest possible, i.e., for the numbers 536 and 214.

Note: For the answer to the first question, it is not necessary to list all possibilities; it is sufficient to argue that the larger number has the hundreds digit 5, and the tens and units digits can be chosen in two ways each, resulting in a total of four possibilities.

If the student concludes that there are eight possibilities because they interpret that it matters which number is written first and which second, they should also receive 2 POINTS for the answer to the first question.

If the student does not mention that we need the largest possible minuend and the smallest possible subtrahend, for the last 3 POINTS, they should calculate the differences \(536-214=322\), \(534-216=318\), \(516-234=282\), and \(514-236=278\) and state the largest, i.e., 322. An answer without reasoning earns 1 POINT.