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# Counting

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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

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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