Finding an equation to a permutation combination question: “Using only three beads, how many different numbers can be made on a three line abacus?”

[Washes cat]

“Using only three beads, how many different numbers can be made on a three line abacus?” I have an answer of 20 different numbers, but not sure what equation to use to obtain the result, or why? I assumed that zero could count as value. I also assumed that the beads could be interchangeable between abacus lines. Please help
Is an answer of 4! - 4 correct?

 

[Dr.Peterson]

“Using only three beads, how many different numbers can be made on a three line abacus?” I have an answer of 20 different numbers, but not sure what equation to use to obtain the result, or why? I assumed that zero could count as value. I also assumed that the beads could be interchangeable between abacus lines. Please help
Is an answer of 4! - 4 correct?

How are you defining "three-line abacus"? In particular, are you modeling this on a Chinese or Japanese type (or something else)? These follow different rules for the number of beads.

Any context you can show for the question will help! Was any picture or explanation provided?

 

[Washes cat]

Thank you for answering. The question is out a book where they’ve given a simple drawing of an abacus that looks like cricket stumps with no bails. The solution involves interchanging beads between the three abacus threads

 

[Dr.Peterson]

Thank you for answering. The question is out a book where they’ve given a simple drawing of an abacus that looks like cricket stumps with no bails. The solution involves interchanging beads between the three abacus threads

Would it be possible to show an image of the entire problem? It still isn't at all clear to me. An abacus doesn't involve interchanging beads.

 

[Harry_the_cat]

I interpret it as "How many 3 digit numbers (including those starting with 00 and 0) can be made where the sum of the digits is three?"

For this interpretation I get 10. (This doesn't include 110, for example, because this doesn't use 3 beads.)

@Washes cat - I think you have interpreted it as "How many three digit numbers (including those starting with 00 and 0) can be made where the sum of the digits is less than or equal to 3?"

Interpreting it this way, I get 20 like you did.

4! - 4 does equal 20, but I can see no logical reason how this expression relates to this problem.

Draw up a tree diagram and you will see that the 20 comes from the sum of the 1st to 4th triangular numbers ie 1+3+6+10.