Chapter 6 - Section 6.5 - Generalized Permutations and Combinations - Exercises - Page 432: 20

364 ways

Given , $x_{1}$+$x_{2}$+$x_{3}$ $\leq$ 11 and all the 3 integers are non-negative. We introduce a auxiliary variable of $x_{4}$ such that $x_{1}$+$x_{2}$+$x_{3}$+$x_{4}$ = 11 and $x_{4}$ $\geq$ 0; $x_{4}$ is an integer Number of ways is given by $C$(n+r-1,r) here n=4 , r=11 So number of ways is given by $C$(14,11) =364 ways