# Using “anti-dots” in stars and bars questions

Find the number of solutions to
$$x_1 + x_2 + \ldots + x_5 = 26$$ where
$x_k \leq 6$ for $k=1,2,\ldots,5$.

This is the solution.
Use the usual stars and bars method except fix every bin with $6$ dots so it looks like
$$......|......|......|......|......$$

Since $30-26=4$, then we want to place $4$ anti-dots into these places so that eahc bins have in total $24$ dots.
It looks like to me that it must be $\binom{5}4$... but the answer has $\binom{8}4$. I don't see how.

• Why do you compute $30-24$? Didn't you mean to compute $30-26$? – lulu Oct 16 '17 at 0:39
• Yes sorry, my bad. I was fixing up the quetsion and forogt that in there. – OneGapLater Oct 16 '17 at 0:50
• Well, using Stars and Bars, you want to count the ways to get $5$ non-negative integers that sum to $4$. That's $\binom {5+4-1}{5-1}=\binom 84$. – lulu Oct 16 '17 at 0:55
• Thanks. But visually we only have 5 spots to place $4$ anti dots in right? Or am I misunderstanding? – OneGapLater Oct 16 '17 at 1:03
• Look up Stars and Bars. The connection to the binomial symbol is spelled out pretty clearly in that article. – lulu Oct 16 '17 at 1:06