The number of unique throw combinations is 48.
The key to solving problems such as this is to break the problem down to a lower level by adopting a logical method of counting, which should ensure that you won't keep counting non-unique combinations of winning throws!
One way is to consider ascending winning combinations only...
- Begin with the lowest number (3) as the 1st score
- Keep on combining all unique winning 2nd & 3rd scores
- Go back to step 1. using next available number (4,5,6....) as 1st score.
For example, look at all combinations involving [3,a,b] where a+b=17, so we've only got two solutions: [3,7,10] & [3,8,9].
Next try [4,c,d], where c+d=16, giving us three solutions: [4,6,10], [4,7,9] & [4,8,8].
Next [5,e,f], where e+f=15, leading to three solutions: [5,5,10], [5,6,9], [5,7,8].
And so on....
We had five correct solutions submitted - the first received was from DEK. Well done!