Consider the Linear Programming problem:Maximize: $7X_1+6X_2+4X_3$ subject to:$X_1+X_2+X_3\le 5$$2X_1+X_2+3X_3\le 10$$X_1,X_2,X_3\ge 0$ The number of basic solutions is:

To find basic solutions, we first convert the inequalities into equations by adding slack variables $S_1$ and $S_2$:$X_1+X_2+X_3+S_1=5$$2X_1+X_2+3X_3+S_2=10$ The total number of variables $n=5$ ($X_1,X_2,X_3,S_1,S_2$).The number of equations $m=2$.The number of possible basic solutions is given by the combination formula:
$$\left(\genfrac{}{}{0ex}{}{n}{m}\right)=\left(\genfrac{}{}{0ex}{}{5}{2}\right)$$
$$\left(\genfrac{}{}{0ex}{}{5}{2}\right)=\frac{5!}{2!(5-2)!}=\frac{5\times 4}{2\times 1}=10$$