Hey guys, can someone please check my solution if its correct for the following question thankyou!!!

QUESTION:

Use inclusion-exclusion to find the number of solutions in non-negative integers to:

with the conditions , , .

MY SOLUTION

STEP 1: Solutions for greater than conditions.

Condition 1:

Condition 2:

Let

Therefore:

So

STEP 2: Solutions for less than conditions.

Condition 1:

Condition 2:

Condition 1:

Let

Therefore:

So

Condition 2:

Let

Therefore:

So

Now we need to find solutions for

Therefore:

NO SOLUTIONS SINCE RHS IS A NEGATIVE INTEGER.......

therefore final solutions: