counting question

11111          

1112   1121  1211  2111

122      212    221

113      131    311

 23         32

The positive integers are, 

11111, 1112, 1121, 1211, 2111, 221, 212, 122, 311, 131, 113, 32, 23.

So, there are 13 numbers that satisfy the condition.

To find the number of positive integers that satisfy both conditions, we can approach it systematically by considering the possible combinations of digits that sum up to 5. Since each digit can be 1, 2, or 3, and we need to find the combinations that sum up to 5, we can start with the smallest possible digit combinations and incrementally increase them until we reach the sum of 5.

Here are the possible combinations:

(1, 1, 1, 1, 1)

(1, 1, 1, 2)

(1, 1, 2, 1)

(1, 2, 1, 1)

(2, 1, 1, 1)

(1, 1, 3)

(1, 3, 1)

(3, 1, 1)

(1, 2, 2)

(2, 1, 2)

(2, 2, 1)

(2, 3)

(3, 2).