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Question

Show that
$$\begin{vmatrix} _{  }^{ x }{ { C }_{ r } } & ^{ x }{ { C }_{ r+1 } } & _{  }^{ x }{ { C }_{ r+2 } } \\ _{  }^{ y }{ { C }_{ r } } & _{  }^{ y }{ { C }_{ r+1 } } & _{  }^{ y }{ { C }_{ r+2 } } \\ _{  }^{ z }{ { C }_{ r } } & _{  }^{ z }{ { C }_{ r+1 } } & _{  }^{ z }{ { C }_{ r+2 } } \end{vmatrix}=\begin{vmatrix} _{  }^{ x }{ { C }_{ r } } & ^{ x+1 }{ { C }_{ r+1 } } & _{  }^{ x+2 }{ { C }_{ r+2 } } \\ _{  }^{ y }{ { C }_{ r } } & _{  }^{ y+1 }{ { C }_{ r+1 } } & _{  }^{ y+2 }{ { C }_{ r+2 } } \\ _{  }^{ z }{ { C }_{ r } } & _{  }^{ z+1 }{ { C }_{ r+1 } } & _{  }^{ z+2 }{ { C }_{ r+2 } } \end{vmatrix}$$


Solution

Hint: First observe that
$$^{ x }{ C }_{ r }+^{ x }{ C }_{ r+1 }=^{ x+1 }{ C }_{ r+1 }$$
and $$^{ x }{ C }_{ r+1 }+^{ x }{ C }_{ r+2 }=^{ x+1 }{ C }_{ r+2 }$$.
Now applying $${C}_{3}+{C}_{2}$$ and $${C}_{2}+{C}_{1}$$, we get
$$\Delta =\left| \begin{matrix} ^{ x }{ C }_{ r } & ^{ x+1 }{ C }_{ r+1 } & ^{ x+1 }{ C }_{ r+2 } \\ ^{ y }{ C }_{ r } & ^{ y+1 }{ C }_{ r+1 } & ^{ y+1 }{ C }_{ r+2 } \\ ^{ z }{ C }_{ r } & ^{ z+1 }{ C }_{ r+1 } & ^{ z+1 }{ C }_{ r+2 } \end{matrix} \right|$$
Again applying $${C}_{3}+{C}_{2}$$, we get 
$$\Delta =\left| \begin{matrix} ^{ x }{ C }_{ r } & ^{ x+1 }{ C }_{ r+1 } & ^{ x+2 }{ C }_{ r+2 } \\ ^{ y }{ C }_{ r } & ^{ y+1 }{ C }_{ r+1 } & ^{ y+2 }{ C }_{ r+2 } \\ ^{ z }{ C }_{ r } & ^{ z+1 }{ C }_{ r+1 } & ^{ z+2 }{ C }_{ r+2 } \end{matrix} \right| \quad $$
by the same rule.

Mathematics

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