The number of solutions to the system of equations `x_1+x_2+x_3+x_4+x_5=20` and `x_1+x_2=15` wher `x_k>=0`

Explanation

We have, `x_(1)+x_(2)+x_(3)+x_(4)+x_(5)=20` . . (i)
and `x_(1)+x_(2)+15` . .. (ii)
Then, from Eqs. (i) and (ii) we get two equations
`x_(3)+x_(4)+x_(5)+5` . .. (iii)
`x_(1)+x_(2)+15`
and given `x_(1) ge0, x_(2)ge0,x_(3)ge0,x_(4)ge0 and x_(5)ge0`
then, number of solutions of Eq. (iii)
`=.^(5+3-1)C_(3-1)=.^(7)C_(2)`
`=(7*6)/(1*2)=21`
and number of solutions of Eq. (iv)
`=.^(15+2-1)C_(2-1)=.^(16)C_(1)=16`
Hence, total number of solutions of the given system of equations `=21xx16=336`