Question

The number of nonnegative integer solutions to the equation \({x_1} + {x_2} + {x_3} = 17\) where \({{\rm{x}}_1},{{\rm{x}}_2},{{\rm{x}}_3}\) are nonnegative integer solutions.

Short answer

The required number of nonnegative integer solutions to the equation \({x_1} + {x_2} + {x_3} = 17\) is 12.

Step-by-step solution

Given data

Equation is \({x_1} + {x_2} + {x_3} = 17\), when \({x_1} < 4,{x_2} < 3,{x_3} > 5\).

Concept used of constraints

Inlinear algebra if the number of constraints (independent equations) in a system of linear equations equals the number of unknowns then precisely one solution exists; if there are fewer independent equations than unknowns, an infinite number of solutions exist; and if the number of independent equations exceeds the number of unknowns, then no solutions exist.

Find the number of solutions

The number of nonnegative integer solutions can be found when the variables are constraints \({x_1} < 6,{x_3} > 5\). A solution to the equation \({x_1} + {x_2} + {x_3} = 17\) subject to these constraints correspondence a selection of 17 items with \({x_1}\) items of one type, \({x_2}\) items of two type and \({x_3}\) items of three types where, in addition, there is at least 6 items of type three together with a choice of additional 11 items of any type is \(C(3 + 11 - 1,11) = C(13,11) = C13,2) = \frac{{13.12}}{{1.2}} = 78\) solutions.

In which we have to subtract the number of solutions for which \({x_1} \ge 4,{x_2} \ge 3\)

\({\rm{ i}}{\rm{.e}}{\rm{., }}C(3 + 7 - 1,7) = C(9,7) = C(9,2) = \frac{{9 \times 8}}{{1 \times 2}} = 36\)

\(C(3 + 8 - 1,8) = C(10,8) = C(10,2) = \frac{{10 \times 9}}{{1 \times 2}} = 45\)

However, there are also solutions in which both restrictions violated i.e., \(C(3 + 4 - 1,4) = C(6,4) = C(6,2) = \frac{{6 \times 5}}{{1 \times 2}} = 15\)

Therefore, the number of solutions in the case of \({x_1} \ge 4\) and \({x_2} \ge 3\) is \(36 + 45 - 15 = 66\) Hence, the required number of nonnegative integer solutions to the equation \({x_1} + {x_2} + {x_3} = 17\) is \(78 - 66 = 12\).