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  • The number of points  in space , whose each  coordinate is a negative integer suc

The number of points (x,y,z) in space , whose each 

coordinate is a negative integer such that x+y+z+12=0 is 

Option: 1

45


Option: 2

55


Option: 3

25


Option: 4

75


Answers (1)

best_answer

\begin{array}{lc} \because & x+y+z=-12 \\ \because & x<0, y<0, z<0 \\ \Rightarrow & -x>0,-y>0,-z>0 \\ \text { or } & -x \geq 1,-y \geq 1,-z \geq 1 \\ \text { Let } & -x=\alpha,-y=\beta,-z=\gamma \end{array} Then, \alpha+\beta+\gamma=12, \alpha \geq 1, \beta \geq 1, \gamma \geq 1\hspace{0.2cm} Then\hspace{0.2cm} number\hspace{0.2cm} of\hspace{0.2cm} points\hspace{0.2cm} $(x, y, z)$ in space \begin{aligned} & ={ }^{12-1} C_{3-1} \\ & ={ }^{11} C_2=\frac{11 \cdot 10}{1}=55 \end{aligned}

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