Find the number of ways of constructing $8$ using three distinct integers from {0,1,2,3,4,5,6,7,8}

Question

In this particular example, order does matter. But at the moment, the only method I can think of is to tediously list them out.

$0+1+7$

$0+2+6$

And so on. But I am thinking that there is surely a better way in general, especially when there are more and more numbers.

Answer 1

Choose 0 as the first number, like you did. Notice that the numbers almost move towards the ends of the set. For instance the 2nd number will get to 7 while the 3rd number will get to 1. So, for each 1st number, you can easily see the number of combinations. Just remember to ignore duplicates.

Answer 2

It is easy enough to list the $5$ ways to do it with $3$ distinct numbers in ascending order, now each of these can be permuted in $6$ different ways, so there are $\color{red}{30}$ ways in toto.