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 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) $w \leq 5$

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