+1234 Find the number of positive integers that satisfy both the following conditions:
- Each digit is a 1 or a 2 or a 3
- The sum of the digits is 5
+1365 We could probably use casework to solve this problem.
The first case to look for is only two digit numbers.
We have 23 and 32, so there are 2 possibilities.
Next, consider 3 digit numbers.
We have 311, 131, 113 for 3 more.
We also have 221, 212, and 122 to add 3 more on.
Next, consider 4 digit numbers.
The only numbers that work are numbers with three 1s and one 2. There are 4 ways we could do this.
Lastly, we have 5 digit numbers. The only one we have to consider is \(11111\), one possibility.
Now, we add up the numbers. We get \(2+3+3+4+1 = 13. \)
So 13 is our answer!
Thanks! :)
