The number of integral solutions to the system the of equations $x_1+x_2+x_3+x_4+x_5=20\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}{x}_1+x_2=15$ when $x_k\ge 0\left(k=1,2,3,4,5\right)$ is

We have, $x_1+x_2+x_3+x_4+x_5=20 ...\left(i\right)$

and       $x_1+x_2=15 ...\left(ii\right)$

Then, from Eqs. (i) and (ii), we get two equations 

             $\begin{array}{l}{x}_3+x_4+x_5=5 ...\left(iii\right)\\ x_1+x_2=15 ...\left(iv\right)\end{array}$

and given $x_1\ge 0,x_2\ge 0,x_3\ge 0,x_1\ge 0\text{ and }{x}_5\ge 0$

Then, number of solutions of Eq. (iii)

                           $\begin{array}{l}{=}^{5+3-1}{C}_{3-1}{=}^7{C}_2\\ =\frac{7\cdot 6}{1\cdot 2}=21\end{array}$

and number of solutions of Eq. (iv) ${=}^{15+2-1}{C}_{2-1}{=}^{16}{C}_1=16$

Hence, total number of solutions of the given system of equations  $=21\times 16=336$