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 $x_1,x_2\dots x_5\in\Bbb{Z^{0+}}\setminus\{3\}$.

The number of solutions of the equation 
$$x_1 + x_2 + x_3 + x_4 + x_5 = 10$$
is the number of ways four addition signs can be inserted into a row of ten ones, which is $\binom{10 + 4}{4} = \binom{14}{4}$.  From these, we must exclude those solutions in which one or more of the numbers is equal to $3$.  Since $3 \cdot 3 = 9 < 10 < 12 = 4 \cdot 3$, at most three of the addends are equal to $3$.  
If one of the addends is equal to $3$, then the sum of the four remaining addends is equal to $10 - 3 = 7$.  The number of solutions of the equation 
$$y_1 + y_2 + y_3 + y_4 = 7$$
in the nonnegative integers is $\binom{7 + 3}{3} = \binom{10}{3}$.  Since there are five ways in which one of the five addends could equal $3$, the number of solutions in which at least one addend is equal to $3$ is $\binom{5}{1}\binom{10}{3}$.  
If two of the addends are equal to $3$, then the sum of the three remaining addends is equal to $10 - 2 \cdot 3 = 4$.  The number of solutions of the equation 
$$z_1 + z_2 + z_3 = 4$$
in the nonnegative integers is $\binom{4 + 2}{2} = \binom{6}{2}$.  Since there are $\binom{5}{2}$ ways in which two of the five addends could equal $3$, the number of solutions in which at least two addends are equal to $3$ is $\binom{5}{2}\binom{6}{2}$.
If three of the addends are equal to $3$, then the sum of the two remaining addends is equal to $10 - 3 \cdot 3 = 1$.  The number of solutions of the equation 
$$w_1 + w_2 = 1$$
in the nonnegative integers is $\binom{1 + 1}{1} = \binom{2}{1}$.  Since there are $\binom{5}{3}$ ways in which three of the five addends could equal $3$, the number of solutions in which three addends are equal to $3$ is $\binom{5}{3}\binom{2}{1}$.  
By the Inclusion-Exclusion Principle, the number of solutions of the equation
$$x_1 + x_2 + x_3 + x_4 + x_5 = 10$$
in which none of the addends is equal to $3$ is 
$$\binom{14}{4} - \binom{5}{1}\binom{10}{3} + \binom{5}{2}\binom{6}{2} - \binom{5}{3}\binom{2}{1} = 531$$
