u | v | w |

4 | 3 | 5 |

4 | 4 | 4 |

4 | 5 | 3 |

4 | 6 | 2 |

5 | 2 | 5 |

5 | 3 | 4 |

5 | 4 | 3 |

5 | 5 | 2 |

6 | 2 | 4 |

6 | 3 | 3 |

6 | 4 | 2 |

7 | 2 | 3 |

7 | 3 | 2 |

8 | 2 | 2 |

Since Here below conditions are satisfied :

1) $v+u+w =12$

2) $u \geq 4$

3) $v , w \geq 2$

4) $w \leq 5$

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no of orange **U** can get is given by $(x^{4}+x^{5}+x^{6}+x^{7}+x^{8})$

no of orange **V **can get is given by $(x^{2}+x^{3}+x^{4}+x^{5})$

no of orange** W** can get is given by $(x^{2}+x^{3}+x^{4}+x^{5}+x^{6})$

Hence finding the coefficient of $x^{12} $ in the equation

$(x^{4}+x^{5}+x^{6}+x^{7}+x^{8})$ * $(x^{2}+x^{3}+x^{4}+x^{5})$ * $(x^{2}+x^{3}+x^{4}+x^{5}+x^{6})$ = $x^{12} $

$x^{4} $( $ \frac{1- x^{5} }{1-x}$) * $x^{2} $( $ \frac{1- x^{4} }{1-x}$) * $x^{2} $( $ \frac{1- x^{5} }{1-x}$) = $x^{8} $ $( 1- x^{5} )*( 1- x^{4} ) *( 1- x^{5} )*({1-x})^{-3}$

Coffiecient of $x^{4 } $ in $( 1- x^{5} )*( 1- x^{4} ) *( 1- x^{5} )*({1-x})^{-3}$ is

$_{6}^{4}\textrm{C}$ - $_{2}^{0}\textrm{C}$ =** 15 -1 =14**