Question

Find the number of non negative integer solutions of $x+2y+3z=30$

Answer 1

Let $x,y,z$ are non- negative integers.

$\begin{array}{c}x>=0,y>=0,z>=0\\ x+2y+3z=0\\ x=30-2y-3z\to \left(i\right)\end{array}$

$10$ possible values of $z$ are $0,1,2,3,4,5,6,7,8,9,10$

$ifz=1,x=30-2y$, $y$ have value from $0,\mathrm{1.......5}$

i.e, $16$ possibilities

$\Rightarrow ifz=1,x=27-2y,y$ have $14$ possibilities

$\Rightarrow ifz=2x=24-2y,y$ have $13$ possibilities

$\Rightarrow ifz=3x=21-2y,y$ have $11$ possibilities

$\Rightarrow ifz=4x=18-2y,y$ have $10$ possibilities

Total possibilities $=16+14+13+11+10+8+7+5+4+2+91$