Advent calendar 2022

There is 1 number whose digits are all non-zero and whose digits add up to 1 (1).

There are 2 numbers whose digits are all non-zero and whose digits add up to 2 (2, and 11).

There are 4 numbers whose digits are all non-zero and whose digits add up to 3 (3, 12, 21, and 111).

There are 8 numbers whose digits are all non-zero and whose digits add up to 4 (4, 13, 31, 112, 121, 211, and 1111).

The amount of numbers is doubling each time. You can justify this by noticing that every number whose digits are all non-zero and whose digits add to \(n+1\) can be made from a number adding to \(n\) by either adding 1 to the final digit or appending a 1 onto the end of the number.

Therefore, there are 128 numbers whose digits are all non-zero and whose digits add up to 8.

There is 1 number whose digits are all non-zero and whose digits add up to 1. There are 2 numbers whose digits are all non-zero and whose digits add up to 2. There are 4 numbers whose digits are all non-zero and whose digits add up to 3. There are 8 numbers whose digits are all non-zero and whose digits add up to 4. There are 16 numbers whose digits are all non-zero and whose digits add up to 5. There are 32 numbers whose digits are all non-zero and whose digits add up to 6. There are 64 numbers whose digits are all non-zero and whose digits add up to 7. There are 128 numbers whose digits are all non-zero and whose digits add up to 8. There are 256 numbers whose digits are all non-zero and whose digits add up to 9.

Are there 512 numbers whose digits are all non-zero and whose digits add up to 10?