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 11.

 Jun 22, 2024

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.

 Jun 22, 2024

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