Question

The number of non negative integral solutions of the equation $x+y+3z=33$ is

• A. $120$
• B. $135$
• C. $210$
• D. $420$

Answer 1

The correct option is C $210$

Consider cases when $z=0,1,2,\dots ,11$
$\Rightarrow x+y=33,30,27,\dots 0$.
$\text{If}{x}_1+x_2=p\text{then no. of solution}={}^{p+2-1}{C}_{2-1}=p+1$

So,Number of solutions when $x+y=33,30,27,\cdots ,0$ will be $(33+1),(30+1),(27+1),\cdots ,(0+1)$ respectively
So, total solutions$=34+31+28+\dots +1$
Lets find number of terms in the above series
$\Rightarrow 34=1+(n-1)3$
$\Rightarrow n=12$
Required number of ways = sum of above series.
$=\frac{12}{2}(1+34)$ $\because [S_n=\frac{n}{2}(a+a_n\left)\right]$
$=210$