Philippine Mathematical Olympiad

Solution:
Using the substitution $(a, b, c, d) = (2a_1 + 1, 2b_1 + 1, 2c_1 + 1, 2d_1 + 1)$ where $a_1, b_1, c_1, d_1$ are nonnegative integers, the problem is equivalent to finding the number $N$ of nonnegative integer solutions of $2a_1 + 1 + 2b_1 + 1 + 2c_1 + 1 + 4d_1 + 2 = 15$ or $a_1 + b_1 + c_1 + 2d_1 = 5$. Note that $d_1 \leq 2$, and for each fixed $d_1 \in \{0, 1, 2\}$, the number of nonnegative integer solutions of $a_1 + b_1 + c_1 = 5 - 2d_1$ is $\binom{7 - 2d_1}{2}$. Hence
$$N = \sum_{d_1 = 0}^{2} \binom{7 - 2d_1}{2} = \binom{7}{2} + \binom{5}{2} + \binom{3}{2} = 21 + 10 + 3 = 34$$