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.

  • 1
    $\begingroup$ Why do you compute $30-24$? Didn't you mean to compute $30-26$? $\endgroup$ – lulu Oct 16 '17 at 0:39
  • $\begingroup$ Yes sorry, my bad. I was fixing up the quetsion and forogt that in there. $\endgroup$ – OneGapLater Oct 16 '17 at 0:50
  • 2
    $\begingroup$ Well, using Stars and Bars, you want to count the ways to get $5$ non-negative integers that sum to $4$. That's $\binom {5+4-1}{5-1}=\binom 84$. $\endgroup$ – lulu Oct 16 '17 at 0:55
  • $\begingroup$ Thanks. But visually we only have 5 spots to place $4$ anti dots in right? Or am I misunderstanding? $\endgroup$ – OneGapLater Oct 16 '17 at 1:03
  • 2
    $\begingroup$ Look up Stars and Bars. The connection to the binomial symbol is spelled out pretty clearly in that article. $\endgroup$ – lulu Oct 16 '17 at 1:06

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