+817 Find the number of positive integers that satisfy both the following conditions:
* Each digit is a $1$ or a $3$ or a $5$
* The sum of the digits is $10$
+6 We can start by making numbers with just the digits. The only ones that work are 1111111111 and 55. Next, we can check for numbers with just 1 and 3:
3331 has 4 placements of 1.
331111 has 15 arrangements.
31111111 has 8 placements of 3.
Then numbers with 1 and 5:
511111 has 6 arrangements.
We can not have numbers with just 3 and 5 so we check for numbers with all 1, 3, and 5. There are 6 possible arrangements.
The answer should be 2+4+15+8+6+6 = 41 such numbers possible.
Please let me know if I got anything wrong.
