Let us look at the given constraints:

x, y, z are non-negative integers =>x>=0,y>=0,z>=0x>=0,y>=0,z>=0

x+y+3z=33=>y=33−3z−x—(1)x+y+3z=33=>y=33−3z−x—(1)

12 possible values for z are 0,1,2,3,4,5,6,7,8,9,10,110,1,2,3,4,5,6,7,8,9,10,11

If z=0, From (1), y = 33 - x => x can have values from 0, 1, .. 33 => 34 possibilities

If z=1, from (1), y = 33 - 3 - x = 30 - x => x can have 31 possibilities ( 0,1,2, .. 30)

If z=2, from (1), y = 33 - 6 - x = 27 - x => x can have 28 possibilities ( 0,1,2, .. 27) and so on

If z=11, from (1) y = 33 - 33 - x = -x => Only one possible combination with x = y = 0

Total possibilities =34+31+28+…+134+31+28+…+1

This is AP with a = 34, d = -3 and n=12

Total possibilities = 34+31+28+…+1=122∗[(2∗34)+(12–1)∗(−3)]=6∗[68−33]=210

**Solution: Number of non-negative integral solutions of the equation x y 3z = 33**

To find the number of non-negative integral solutions of the equation x y 3z = 33, we can use the concept of prime factorization and counting techniques.

**Step 1: Prime factorization of 33**

We can write 33 as a product of its prime factors as follows:

33 = 3 x 11

**Step 2: Counting the number of solutions**

Now, we need to count the number of non-negative integral solutions of the equation x y 3z = 33.

Let's consider each prime factor separately and count the number of solutions for each.

For the prime factor 3, we can have the following cases:

- x = 0, y = 0, z = 11 (1 solution)
- x = 1, y = 0, z = 11 (2 solutions)
- x = 2, y = 0, z = 11 (2 solutions)
- x = 0, y = 3, z = 3 (4 solutions)
- x = 1, y = 1, z = 3 (4 solutions)
- x = 0, y = 6, z = 1 (2 solutions)

Total number of solutions for the prime factor 3 = 15

Similarly, for the prime factor 11, we can have the following cases:

- x = 0, y = 0, z = 3 (1 solution)

Total number of solutions for the prime factor 11 = 1

Therefore, the total number of non-negative integral solutions of the equation x y 3z = 33 is equal to the product of the number of solutions for each prime factor, which is:

Total number of solutions = 15 x 1 = 15

Hence, there are 15 non-negative integral solutions of the equation x y 3z = 33.