Find the number of non-negative intergral solutions of `x_(1)+x_(2)+x_(3)+x_(4)=20`.

Explanation

We have `x_(1)+x_(2)+x_(3)+4x_(4)=20`, where `x_(1),x_(2),x_(3),x_(4) ge 0`.
If we let `4x_(4)=y_(4)` we have
`x_(1)+x_(2)+x_(3)+y_(4)=20, " where" y_(4)=0,4,8,12,16 " or " 20`.
`therefore` Number of non-negative integral solutions of the above equation
=coefficient of `p^(20) " in " underset("for" x_(1),x_(2) "and " x_(3))ubrace((p^(0)+p^(1)+p^(2)+p^(3)+..)^(3))xxunderset("for" y_(4))ubrace((p^(0)+p^(4)+p^(8)+p^(12)+p^(16)+p^(20)))`
=coefficient of `p^(20) " in " ((1)/(1-p))^(3)((1)/(1-p^(4)))`
=coefficient of `p^(20) " in " (1-p)^(-3)(1-p^(4))^(-1)`
=coefficient of `p^(20) " in " (1+ ""^(3)C_(1)p+ ""^(4)C_(2)p^(2)+ ""^(5)C_(3)p^(3)+ ""^(6)C_(4)p^(4)+..) (1+p^(4)+p^(8)+..)`
`=1+ ""^(6)C_(4)+""^(10)C_(8)+ ""^(14)C_(12)+ ""^(18)C_(16)+""^(22)C_(20)=536`