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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\$ or a \$5\$
The sum of the digits is \$5\$

Dec 16, 2023

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Here's how to find the number of positive integers that satisfy the conditions:

Build the numbers systematically:

Consider the number of digits: We need a two-digit or three-digit number since the sum of digits is 5.

Two-digit numbers:

With two digits, there are three choices for the first digit (1, 3, or 5) and two choices for the second digit. This gives us 3 * 2 = 6 possibilities.

However, these possibilities include leading zeros (e.g., 13, 31), which are not positive integers. We need to remove them.

Three-digit numbers:

With three digits, there are three choices for each digit, giving us 3 * 3 * 3 = 27 possibilities.

None of these will have leading zeros since the first digit can also be 1, 3, or 5.

Remove leading zeros from two-digit numbers:

There are only two leading zeros possible (10 and 30), so we subtract them from the two-digit possibilities:

Number of valid two-digit numbers = 6 - 2 = 4

Combine possibilities and find the total:

Adding the valid two-digit and three-digit possibilities:

Total number of positive integers = 4 + 27 = 31

Therefore, there are 31 positive integers that satisfy both conditions.

Dec 17, 2023