Total number of integral solutions of the system ,equations  $x_1+x_2+x_3+x_4+x_5=20\text{ and }{x}_1+x_2+x_3=5$ when $x_k\ge 0$ is

We have,

and $\begin{array}{l}{x}_1+x_2+x_3+x_4+x_5=20\\ {\ddot{x}}_1+x_2+x_3=5\end{array}$

These two equations reduce to

and $\begin{array}{r}{x}_4+x_5=15\\ x_1+x_2+x_3=5\end{array}$

Since corresponding to each solution of (i) there are solutions of equation (ii). So, total number of solutions of the given system of equations 

= No. of solutions of (i) x No. of solutions of (ii)

${=}^{15+2-1}{C}_{2-1}{\times }^{5+3-1}{C}_{3-1}{=}^{16}{C}_1{\times }^7{C}_2=336$