3. (10 points $=5+5) Consider the expression a+b+c=7 ,where a, b, and c are nonnegative integers.
a. How many possible solutions are there?

Step 1 ：This is a combinatorics problem. We are looking for the number of ways to distribute 7 identical items (representing the value 7) into 3 distinct boxes (representing the variables a, b, and c). This is a problem of combinations with repetition.

Step 2 ：This can be solved using the formula for combinations with repetition: \(C(n + r - 1, r) = C(n + r - 1, n - 1)\), where n is the number of items (7 in this case) and r is the number of boxes (3 in this case).

Step 3 ：Substituting the given values into the formula, we get \(C(7 + 3 - 1, 3) = C(9, 3)\).

Step 4 ：Calculating the combinations, we find that there are 36 possible solutions.

Step 5 ：Final Answer: There are \(\boxed{36}\) possible solutions.