Question #40598

Let $a$ be the number of ways to select 4 items from 3 distinct types of items (assuming an unlimited supply of each type). Let $b$ be the number of ways to distribute 5 identical balls into 3 distinct boxes such that each box receives at least 1 ball. Let $c$ be the number of ways to select 2 items from 4 distinct types of items (assuming an unlimited supply of each type). Consider the system of linear equations: $\begin{aligned} a_{11}x + a_{12}y + a_{13}z &= d_1 \\ a_{21}x + a_{22}y + a_{23}z &= d_2 \\ a_{31}x + a_{32}y + a_{33}z &= d_3 \end{aligned}$ where the coefficient matrix $M = [a_{ij}]$ is a skew-symmetric matrix with $a_{12} = a$, $a_{13} = b$, and $a_{23} = c$. If the system has infinitely many solutions, the vector $\vec{d} = d_1 \hat{i} + d_2 \hat{j} + d_3 \hat{k}$ must be orthogonal to which of the following vectors?