Find the number of positive integral solutions of the inequality `3x+y+zlt=30.`

Explanation

Let dummy variable w, then,
`3x+y+z+w=30,wge0` . .. (i)
now, let `a=x-1,b=y-1,c=z-1,d=w`, then
`3a+b+c+d=25,` where a,b,c,d`ge0` . . (ii)
`therefore`Number of positive integral solutions of Eq. (i)
=Number of non-negative integral solutions of Eq. (ii)
=Coefficient of `alpha^(25)` in `(1+alpha^(3)+alpha^(6)+ . ..+(1+alpha+alpha^(2)+ . .)^(3)`
=Coeffieint of `alpha^(25)` in `(1+alpha^(3)+alpha^(6)+ . . .)(1-alpha)^(-3)`
=Coefficient of `alpha^(25)` in
`(1+alpha^(3)+alpha^(6)+ . .)(1+.^(3)C_(1)alpha+.^(4)C_(2)alpha^(2)+ . . )`
`=.^(27)C_(25)+.^(24)C_(22)+.^(21)C_(19)+.^(18)C_(16)+.^(15)C_(13)+.^(12)C_(10)+.^(9)C_(7)+.^(6)C_(4)+.^(3)C_(1)`
`=.^(27)C_(2)+.^(24)C_(2)+.^(21)C_(2)+.^(18)C_(2)+.^(15)C_(2)+.^(12)C_(2)+.^(9)C_(2)+.^(6)C_(2)+.^(3)C_(1)`
`=351+276+210+153+105+66+36+15+3=1215`.