Finding Nash Equilibrium for 2x3 game *with no Pure NE*

Question

There's no pure NE nor dominated strategies and I am struggling to solve for the MSNE when 2x3 matrix like this... $$\begin{array}{|c|c|c|c|}\hline & C & D & E \\ \hline A & 0,10 & 10,0 & -5,7\\ \hline B & 10,0 & 0,10 & -5,7 \\ \hline\end{array}$$

Answer 1

Suppose player $1$ chooses $A$ with probability $p$, and $2$ chooses $C$ and $D$ with probability $q$ and $s$ respectively. Then $E(\pi_2)=10qp+10s(1-p)+7(1-q-s)$, and solving the first order conditions yields that a mixed strategy equilibrium must satisfy $p=3/10$ or $p=7/10$. In both cases, option $E$ weakly dominates options $C$ and $D$ for player $2$, so any strategy where player $1$ chooses $A$ with $p\in[3/10,7/10]$ and player $2$ chooses $E$ with probability $1$, is a mixed strategy Nash equilibrium.