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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 $10$

 Oct 29, 2023
 #1
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We can start by making numbers with just the digits. The only ones that work are 1111111111 and 55. Next, we can check for numbers with just 1 and 3:

3331 has 4 placements of 1.

331111 has 15 arrangements.

31111111 has 8 placements of 3.

 

Then numbers with 1 and 5:

511111 has 6 arrangements.

 

We can not have numbers with just 3 and 5 so we check for numbers with all 1, 3, and 5. There are 6 possible arrangements.

 

The answer should be 2+4+15+8+6+6 = 41 such numbers possible.

 

Please let me know if I got anything wrong.

 Oct 29, 2023

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