2011 Pascal Problem 23

An ordered list of four numbers is called a *quadruple.* A quadruple (p, q, r,\ \text{s}) of integers with p, q, r,\ \text{s} \ge 0 is chosen at random such that 2p + q + r + s = 4 What is the probability that p + q + r + s = 3?

\textbf{(A)}\ \frac{3}{22}\quad \textbf{(B)}\ \frac{3}{11}\quad \textbf{(C)}\ \frac{3}{19}\quad \textbf{(D)}\ \frac{6}{19}\quad \textbf{(E)}\ \frac{2}{7}

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Pascal contests here.