Find the number of positive unequal integral solutions of the equation `x+y+z+w=20`.

Explanation

We have, `x+y+z+w=20` . . (i)
Assume `x lt y lt z lt w.` here, x,y,z,w`ge1`
Now, let `x=x_(1),y-x=x_(2),z-y=x_(3) and w-z=x_(4)`
`thereforex=x_(1),y=x_(1)+x_(2),z=x_(1)+x_(2)+x_(3)` and
`w=x_(1)+x_(2)+x_(3)+x_(4)`
from eq. (i), `4x_(1)+3x_(2)+2x_(3)+x_(4)=20`
then, `x_(1),x_(2),x_(3),x_(4)ge1`.
`because4x_(1)+3x_(2)+2x_(3)+x_(4)=20` . .. (ii)
`therefore`Number of positive integral solution of Eq. (ii)
=Coefficient of `x^(20-10)` in
`(1-x^(4))^(-1)(1-x^(3))^(-1)(1-x^(2))^(-1)(1-x)^(-1)`
=Coefficient of `x^(10)` in
`(1-x^(4))^(-1)(1-x^(3))^(-1)(1-x^(2))^(-1)(1-x)^(-1)`
=Coefficient of `x^(10)` in `(1+x^(4)+x^(8)+x^(12)+ . ..)xx(1+x^(3)+x^(6)+x^(9)+x^(12)+ . .. )xx(1+x^(2)+x^(4)+x^(6)+x^(8)+x^(10)+ . ..)xx(1+x+x^(2)+x^(3)+x^(4)+x^(5)+x^(6)+x^(7)+x^(8)+x^(9)+x^(10)+ . ..)`
=Coefficient of `x^(10)` in
=Coefficient of `x^(10)` in
`(1+x^(3)+x^(6)+x^(9)+x^(4)+x^(7)+x^(10)+x^(8))xx(1+x^(2)+x^(4)+x^(6)+x^(8)+x^(10))(1+x+x^(2)+x^(3)+x^(4)+x^(5)+x^(6)+x^(7)+x^(8)+x^(9)+x^(10))` [neglecting higher powers]
=Coefficient of `x^(10)` in
`(1+x^(2)+x^(4)+x^(6)+x^(8)+x^(10)+x^(3)+x^(5)+x^(7)+x^(9)+x^(6)+x^(8)+x^(10)+x^(9)+x^(4)+x^(6)+x^(8)+x^(10)+x^(7)+x^(8)+x^(10)+x^(8)+x^(10))(1+x+x^(2)+x^(3)+x^(4)+x^(5)+x^(6)+x^(7)+x^(8)+x^(9)+x^(10))` [neglecting higher power]
`=1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1=23`
But x,y,z and w can be arranged in `.^(4)P_(4)=4!=24`
Hence, required number of sols `=(23)(24)=552`