Find the number of integer solutions to the following equation: $$x_1+x_2+\cdots+x_6=12$$
where $x_1, x_5\ge 0$ and $x_2, x_3, x_4 > 0$
This problem can be transformed to a case of counting all positive integer solutions or counting all non-negative integer solutions.
Solution 1
Let $y_1 = x_1 +1$ and $y_5 = x_5+1$, Then the original equation becomes $$y_1 + x_2 + x_3 + x_4 + y_5 = 14$$
where all the variables are positive integers. This is a basic pattern and the answer is $$\binom{14-1}{5-1}=\boxed{715}$$
Solution 2
Let $y_2 = x_2 -1, y_3 = x_3 -1$, and $y_4=x_4-1$. Then the original equation becomes $$x_1 + y_2 + y_3 + y_4 + x_5 = 9$$
where all variables are non-negative integers. This is a basic pattern and the answer is $$\binom{9+(5-1)}{5-1}=\boxed{715}$$
These two answers agree with each other.