Find the number of positive integers that satisfy both the following conditions:
- Each digit is a \(1\) or a \(3\)
- The sum of the digits is \(12\)
I have a vague idea that this is solved with casework, but I don't know where to start... please help!
We have 8 slots for the digits of our number, and we need to place a 1 in 4 of those slots. This can be done in C(8,4) ways.
Once we have placed the 1's, we can fill in the remaining slots with 3's. This can be done in 4! ways.
Therefore, the total number of positive integers that satisfy both of the given conditions is C(8,4)*4! = 416.