<s>[INST] The number of ways to write $10$ as the sum of five natural numbers not equal to $3$ How many answers are there for the equation
$$x_1+x_2+x_3+x_4+x_5=10$$
given that $x_1,x_2\dots x_5\in\Bbb{Z^{0+}}\setminus\{3\}$.
 [/INST] What's easy to compute is we know $j$ of them is equal to $3$, say the result is $S_j$
$$S_j=\binom{5}{j}\binom{10-3j+(5-j-1)}{(5-j-1)}=\binom{5}{j}\binom{14-4j}{4-j},$$
notice that $0\le j\le 3.$ And that $(10-3j)$, $(5-j-1)$ are what called Stars, Bars respectively. By exactly-IEP we have
$$\begin{align}E_0&=\sum_{j=0}^{3}(-1)^j\binom{j}{0}S_j\\
&=\sum_{j=0}^3(-1)^j\binom{5}{j}\binom{14-4j}{4-j}\\
&=+\binom{14}{4}-\binom{5}{1}\binom{10}{3}+\binom{5}{2}\binom{6}{2}-\binom{5}{3}\binom{2}{1}\\
&=531.\quad\quad\square
\end{align}$$
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