1

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.

Trogdor
  • 10,331
  • Start from the largest value, $8$. How many ways can you make $8$ using that? Next, consider $7$: How many ways can you make $8$ using that $7$? And so on. – David G. Stork Sep 18 '18 at 00:05
  • Hint: stars and bars. – Barry Cipra Sep 18 '18 at 00:13
  • 1
    @BarryCipra I thought of the stars and bars method. But how can we ensure that the three integers are distinct? We can construct 8 stars and add in two bars, but what is to prevent a $1+1+6$ scenario occurring? – Trogdor Sep 18 '18 at 00:15
  • @Trogdor, ah, you're right. I missed that crucial condition. Sorry! – Barry Cipra Sep 18 '18 at 00:17

2 Answers2

1

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.

Donald Splutterwit
  • 36,613
  • 2
  • 26
  • 73
0

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.

PreciselyT
  • 3
  • 3
  • Or just use a computer: Mathematica instantly gives ${{8},{7,1},{6,2},{6,1,1},{5,3},{5,2,1},{5,1,1,1},{4,4},{4,3,1},{4,2,2},{4,2,1,1},{4,1,1,1,1},{ 3,3,2},{3,3,1,1},{3,2,2,1},{3,2,1,1,1},{3,1,1,1,1,1},{2,2,2,2},{2,2,2,1,1},{2,2,1,1,1,1},{2,1,1,1,1,1 ,1},{1,1,1,1,1,1,1,1}}$ – David G. Stork Sep 18 '18 at 00:11
  • 3
    @DavidG.Stork, I think you're solving a different problem than what the OP is asking. – Barry Cipra Sep 18 '18 at 00:13