Counting

We can solve this problem by considering the number of 1s and 3s used to form the sum 11.

 

Number of 1s and 3s:

 

The sum of the digits is 11, which can be formed using digits 1 and 3. Let's say there are x ones and (11 - x) threes.

 

Restrictions on x:

 

Since each digit must be a 1 or a 3, x (number of ones) can range from 0 (all digits are threes) to 11 (all digits are ones). However, we only care about positive integers that satisfy the condition. Therefore, valid values for x are:

 

1 <= x <= 11

 

Counting the Possibilities:

 

For each value of x (number of ones), we have (11 - x) threes. So, the total number of positive integers with x ones and (11 - x) threes is 1.

 

This creates 11 possibilities (from x = 1 to x = 11) since there are 11 valid values for x.

 

Therefore, there are 11 positive integers that satisfy both conditions.