whats wrong with this 00 Let's think step by step. The smallest three-digit number possible by removing a three-digit number formed using the digits 8, 5, 4, 3, 2, and 1 exactly once from another three-digit number formed using the same digits exactly once is 100. One way to approach this problem is to start by listing all the possible three-digit numbers that can be made using the digits 8, 5, 4, 3, 2, and 1 exactly once. There are 6!/(3!3!) = 20 of them: 123, 124, 125, 128, 132, 134, 135, 138, 142, 143, 145, 148, 152, 153, 154, 158, 182, 183, 184, 185 Next, let's consider what happens when we remove one of these numbers from another one. We want to minimize the resulting number, so we should try to remove the largest number possible. The largest three-digit number in the list is 854. If we remove any other three-digit number from it, we will get a number that is still three-digit. For example, removing 123 gives 731, which is three-digit. However, if we remove 854 from itself, we get 0, which is not three-digit. So 854 is not a valid choice for the number that we need to remove. The next largest three-digit number in the list is 845. If we remove any other three-digit number from it, we will get a number that is still three-digit. For example, removing 123 gives 722, which is three-digit. However, if we remove 845 from itself, we get 0, which is not three-digit. So 845 is not a valid choice for the number that we need to remove. We can continue this process with the next largest three-digit numbers in the list: 835, 825, 815, 548, 538, 528, 518, 458, 438, 428, 418, 358, 348, 328, 318, 258, 248, 238, 218. For each of these numbers, if we remove any other three-digit number from it, we will get a number that is still three-digit. However, if we remove the number itself, we will get 0, which is not three-digit. So none of these numbers are valid choices for the number that we need to remove. This means that the smallest three-digit number that we can make by removing a three-digit number formed using the digits 8, 5, 4, 3, 2, and 1 exactly once from another three-digit number formed using the same digits exactly once must be one of the smallest three-digit numbers in the list: 123, 124, 125, 128, 132, 134, 135, 138, 142, 143, 145, 148, 152, 153, 154, 158, 182, 183, 184, 185. Comparing these numbers, we see that the smallest one is 123. So the smallest three-digit number possible by removing a three-digit number formed using the digits 8, 5, 4, 3, 2, and 1 exactly once from another three-digit number formed using the same digits exactly once is 123. Therefore, the answer is 123.