How many integral solutions are there to the system of equations `x_(1)+x_(2)+x_(3)+x_(4)+x_(5)=20 and x_(1)+x_(2)+x_(3)=5` when `x_(k) ge 0`?

Options

A. 335
B. 336
C. 338
D. 340

Explanation

We have, `x_(1)+x_(2)+x_(3)+x_(4)+x_(5)=20and x_(1)+x_(2)+x_(3)=5`
These two equtions reduce to
`x_(4)+x_(5)=15" "(i)`
and `x_(1)+x_(2)+x_(3)=5" "(ii)`
Since corresponding to each solution of (i) there are solutions of equation (ii). So, total number of solution of the given system of equations
= No. of solutions of (i) `xx` No. of solutions of (ii)
`=""^(15+2-1)C_(2-1)xx""^(5+3-1)C_(3)=""^(16)C_(1)xx""^(7)C_(2)=336`.