(a) Find the non-negative integral solutions of the equation `x+y+z+t=15`,
(b) Find the number of non-negative integral solutions `x_(1)+x_(2)+x_(3)+4x_(4)=20`

Explanation

(a) The required number of non-negative integral solution of the given equation `x+y+z+t=14`
is equal to the coefficient of `x^(15)` in `(x_(0)+x_(1)+x_(2)+x_(3)+….)^(4)`
=Coeff. Of `x^(15)` in `(1-x)^(-4)`
=Coeff. Of `x^(15)` in `(1+""^(4)C_(1)x + ""^(5)C_(2)x^(2)+""^(6)C_(3)x^(3)+...+""^(18)C_(15)x^(15)+...)`
`=""^(18)C_(15)=(18!)/(15!3!)=(18xx17xx16)/(6)=51xx16=816`
(b) The given equation is `x_(1)+x_(2)+x_(3)+4x_(4)` =20, Here `X_(1),X_(2),X_(3),X_(4) ge 0`
Keeping `x_(4)` fixed, say K i.e., `x_(4)=k` where `0 le k le 5` , the number of non-negative integral solution to the equation ltbRgt `x_(1)+x_(2)+x_(3)=20-4x_(4)=20-4k`
is `""^(20-4k+3-`)C_(3-1)=""^(22-4k)C_(2)`
Where `0 le k le 5`
Hence , the required number of non-negative integral solution of the given equation `x_(1)+x_(2)+x_(3)+4x_(4)=20`
`underset(k=0)overset(5)sum ""^(22-4k)C_(2)=""^(22)C_(2)+""^(18)C_(2)+""^(14)C_(2)+""^(10)C_(2)+""^(6)C_(2)+""^(2)C_(2)`
`=231+153+91+45+15+1=536`