Question

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

Answer 1

Hint: We will first rearrange the given equation. Then we will see all possible values for one variable and then look for all possible solutions for the remaining two variables. We will look for a pattern in the number of solutions after fixing one variable. Then we will add the number of all possible solutions.

Complete step-by-step answer:
The given equation is $x+2y+3z=30$. Let us shift the $z$ variable to the other side of the equation. We get the following equation, $x+2y=30-3z$. We want to find the number of all possible non-negative integer solutions. So, from the equation it is clear that the variable $z$ can take values from $0,1,\dots 10$.
Now, let us fix $z=0$. The possible values for $(x,y)$ which satisfy the equation are $(0,15),(2,14),(4,13),\dots (28,1),(30,0)$. These are 16 solutions.
Next, let us fix $z=1$. The solutions for the equation $x+2y=27$ are $(1,13),(3,12),(5,11),\dots (25,1),(27,0)$. These are 14 solutions.
Fixing $z=2$, the equation becomes $x+2y=24$. The possible solutions are $(0,12),(2,11),\dots ,(24,0)$. These are 13 solutions.
For the next values of $z=3,\dots ,10$, the number of solutions turn out to be 11, 10, 8, 7, 5, 4, 2 and 1 respectively.
So, the total number of non-negative integer solutions for the given equation is $1+2+4+5+7+8+10+11+13+14+16=91$.
The number of non negative integer solutions of $x+2y+3z=30$ is $91$.

Note: We can see a pattern in the number of solutions as we change the value of $z$. We see that these numbers are all positive numbers from 1 to 16, except for the multiples of 3. So, we can calculate the total number by adding the first 16 positive numbers and subtracting the sum of the multiples of 3 that are less than 16. This means that the total number is $1+2+\cdots +16-\sum _{n=1}^{5}3n={\displaystyle \frac{16\times 17}{2}}-45=136-45=91$.