Find the number of positive integers that satisfy both the following conditions:

Each digit is a $1$ or a $2$ or a $3$

The sum of the digits is $5$

BRAlNBOLT May 24, 2024

#1**0 **

We can use constructive counting to solve this problem.

**CASE 1**

First, let's see all the two digit numbers that satisfy these conditions.

We only have two, with 23 and 32.

**CASE 2**

Now, let's check all the 3 digit numbers.

We have 311, 131, 113, giving us 3.

We also have 221, 212, and 122, giving us another 3.

**Case 3**

4 digit number time! This is where it gets a bit complicated, but it's still not that bad.

We have numbers with three 1s and one 2, which has 4 different possibilities.

**Case 4**

Last case ! 5 digit numbers only have 1, with 11111.

Now, we just have to add up all our cases, and we get 2+3+3+4+1 = 13.

We have 13 numbers that satisfy the conditions.

NOTICE: This problem didn't have a lot of cases, so it would be easy to count. Bigger and wider ranges would require permutations!

Thanks! :)

NotThatSmart May 24, 2024