Find the number of non-negative integral solutions of `x_1+x_2+x_3+4x_4=20.`

Explanation

Number of non-negative integral solutions of the given equation
=Coefficient of `x^(20)` in `(1-x)^(-1)(1-x)^(-1)(1-x)^(-1)(1-x^(4))^(-1)`
=Coefficient of `x^(20)` in `(1-x)^(-3)(1-x^(4))^(-1)`
=Coefficient of `x^(20)` in `(1+.^(3)C_(1)+.^(4)C_(2)x^(2)+.^(5)C_(3)x^(3)+.^(6)C_(4)x^(4)+ . ..+.^(10)C_(8)x^(8)+ . .+.^(14)C_(12)x^(12)+ . . +.^(18)C_(16)x^(16)+ . . .+.^(22)C_(20)x^(20)+ . .. )(1+x^(4)+x^(8)+x^(12)+x^(16)+x^(20)+ . .)`
`=1+.^(6)C_(4)+.^(10)C_(8)+.^(14)C_(12)+.^(18)C_(16)+.^(22)C_(20)`
`=1+.^(6)C_(2)+.^(10)C_(2)+.^(14)C_(2)+.^(18)C_(2)+.^(22)C_(2)`
`=1+((6*5)/(1*2))+((10*9)/(1*2))+((14*13)/(1*2))+((18*17)/(1*2))+((22*21)/(1*2))`
`=1+15+45+91+153+231=536`