3

I have a set of numbers $i=1,...,100$.

How many combinations exist using numbers from this set that sum to 100 of length 8?

So for example these would be valid solutions:

$(1, 2, 3, 4, 5, 6, 7, 72)$, $(10,11,22,1,5,8,9,34)$

also, the order is important, that means $(1, 2, 3, 4, 5, 6, 7, 72) \neq (2,1, 3, 4, 5, 6, 7, 72)$

also zeros can be added in any amount, e.g. $(100,0,0,0,0,0,0,0)$.

numbers can also be used more than one time, e.g.

$(10,10,10,10,10,10,10,30)$

Peter Taylor
  • 13,425
spore234
  • 217

1 Answers1

2

If $(100,0,0,0,0,0,0,0) \neq (0,100,0,0,0,0,0,0)$, for instance, you're looking for the number of solutions $(x_1, x_2, ..., x_8)$ to the equation $x_1 + ... + x_8 = 100$, where each $x_i$ is a non-negative integer.

The solution is given by a stars-and-bars argument. In your particular case, the answer is $$ \binom{107}{7}. $$

Otherwise, if the zeroes can only be added at the end, you're looking for the number of solutions to $x_1 + ... + x_k = 100$, where $k \le 8$ and each $x_i$ is a positive integer. A similar stars-and-bars argument gives $$ \sum_{k=1}^8 \binom{99}{k-1}. $$

Shagnik
  • 3,613