How many seven-digit numbers less than 1,048,000 can be made using the digits 0,1,4,6,7,8, and 9? Make a list of possible ways to represent your seven-digit numbers.

The constraints are:

1. the number has to have 7 digits (i.e. the first digit cannot be zero)
2. the resulting number cannot be more than 104800.

Starting from the left-most digit, ask yourself which digits will fit the constraints.

Continue this way for the remaining digits. If there is more than one choice, make a list of each possibility, and work on each separately.

2x9x9x9x9x9x9

To determine the number of seven-digit numbers that can be made using the given digits, we need to consider the restrictions on each digit's position.

Since there are no restrictions on the leading digit, it can be any of the seven given digits (0, 1, 4, 6, 7, 8, or 9).

For the second digit, we can use any of the seven given digits, as there are no restrictions.

For the third digit, we can only use digits that are smaller than 8, as the number needs to be less than 1,048,000. Therefore, we have six options (0, 1, 4, 6, 7, or 8).

For the fourth digit, again, we have six options (0, 1, 4, 6, 7, or 8).

For the fifth digit, we are restricted to three options (0, 1, or 4), as the number has to be less than 1,048,000.

For the sixth digit, again, we have three options (0, 1, or 4).

For the seventh digit, we are restricted to four options (0, 1, 6, or 7).

To find the total number of possible seven-digit numbers under the given conditions, we multiply the number of options for each digit position together:

7 (options for the first digit) × 7 (options for the second digit) × 6 (options for the third digit) × 6 (options for the fourth digit) × 3 (options for the fifth digit) × 3 (options for the sixth digit) × 4 (options for the seventh digit) = 2,7648.

So, there are 27,648 seven-digit numbers that can be made using the given digits.

As for providing the list of possible ways to represent the seven-digit numbers, it is not feasible to manually list all 27,648 combinations. However, you can generate the numbers programmatically using a computer program or use mathematical techniques if you specifically want certain patterns or conditions in the numbers.