<s>[INST] Number of positive integral solutions to $x+y+z+w=20$ with $xWhat is the number of positive unequal integral solution of the equation $x+y+z+w=20$, if $\,x<y<z<w\,$ and $\,x,y,z,w\ge1\;?$ 
How to solve this question?
 [/INST] There are following posible solution:
1,2,3,14  $\,\,\,\,\,\, $    1,3,4,12,  $\,\,\,\,\,\, $ 1,4,5,10  $\,\,\,\,\,\, $ 1,5,6,8
1, 2,4,13   $\,\,\,\,\,\, $   1,3,5,11   $\,\,\,\,\,\, $ 1,4,6,9
1,2,5,12   $\,\,\,\,\,\, $    1,3,6,10     $\,\,\,\,\,\, $ 1,4,7,8
1,2,6,11    $\,\,\,\,\,\, $   1,3,7,9
1,2,7,10
1,2,8,9
So there is 14 $\times $ 3 = 42 if $x=1$
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