How many solutions are there to the equation below, if $x_i$ is a positive integer > 1:
$$\sum_{1}^{6} x_i = 29$$
I also have to do this for x1<=5, however I imagine that's a similar process.
So, my question is based on $x_i>1$. Where is a good starting point?
Edit: I feel as though I should use bars and stars in some way. We have 6 bars and 17 stars, then multiply that by the possible number of ways that we can order the bars (i.e. $x_1,x_2,x_3,x_4,x_5,x_6 \space \text{vs} \space x_1,x_2,x_3,x_4,x_5,x_6$)?
After realizing I can have a min of 2 and max of 19 what's a step in the right direction?
Also, I just realized I've had the idea in my mind that 2 + 2 + 2 + 2 + 2 + 19 and 19 + 2 + 2 + 2 + 2 + 2 would be different solutions, but I now believe those would count as the same. Just as some clarification, plus to explain why I wanted to multiply by 6! in my previous attempt
Best Answer
We can also use the Stars and bars argument -- associate with each $x_i$ the variable $y_i$ such that $x_i = 2 + y_i$. Then, we have an equivalent problem in:
$$\sum_{i=0}^6 y_i = 17$$
(subject to $y_i \ge 0$). Using the stars and bars argument we immediately have $\binom{22}{5}$ as our answer.