in Combinatory edited by
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Answer given is 14

in Combinatory edited by
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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) $w \leq 5$ 

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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