Integer partitions of 12 given odd and even conditions to the components.

Question

How many integer solutions are there to the equation $x_1 + x_2 + x_3 + x_4=12$ with $x_1$ and $x_2$ odd and $x_3$ and $x_4$ even?

So I know that for the integer $12$ there are $p(12)=77$ partitions and I have also deduced there are $15$ partitions of the integer $12$ of size $4$. Where I'm stuck is determining how many of these have $x_1$ and $x_2$ odd and $x_3$ and $x_4$ even.

Answer 1

Subtract $1$ from the last two, giving four even numbers that sum to $10$. Divide them all by $2$, giving four numbers that sum to $5$. Do you know how to solve that?