How many seven digit numbers less than 1.048,000 can be made using the digits, 0,1,4,6,7,8,9? Make a list of possible ways to represent your seven digit numbers
10
1,047,000. 1,046,000
2x1x1x1x1x1
To find the number of seven-digit numbers that can be made using the given digits and are less than 1,048,000, we need to consider the restrictions on each digit's placement.
1. The leftmost digit must be either 0, 1, or 4 to ensure the number is less than 1,048,000.
2. The remaining six digits can be chosen from any of the given digits: 0, 1, 4, 6, 7, 8, 9.
Let's generate the list of possible seven-digit numbers:
1. With 0 as the leftmost digit:
- 0 _ _ _ _ _ _
- The remaining six digits can be any of the given digits: 1, 4, 6, 7, 8, 9.
- Therefore, there are 6 choices for each of the remaining six digits.
- The total number of numbers with 0 as the leftmost digit is 6^6 = 46656.
2. With 1 as the leftmost digit:
- 1 _ _ _ _ _ _
- The remaining six digits can be any of the given digits: 0, 1, 4, 6, 7, 8, 9.
- Therefore, there are 7 choices for each of the remaining six digits.
- The total number of numbers with 1 as the leftmost digit is 7^6 = 117,649.
3. With 4 as the leftmost digit:
- 4 _ _ _ _ _ _
- Since the number needs to be less than 1,048,000, the next digit cannot be greater than 0.
- Therefore, there is only 1 choice for the next digit (0).
- The remaining five digits can be any of the given digits: 1, 4, 6, 7, 8, 9.
- Therefore, there are 6 choices for each of the remaining five digits.
- The total number of numbers with 4 as the leftmost digit is 1 * 6^5 = 7776.
To obtain the final total count, sum up the results from each case:
46656 + 117649 + 7776 = 171,081
So, there are 171,081 possible seven-digit numbers less than 1,048,000 that can be made using the given digits (0, 1, 4, 6, 7, 8, 9).