Number of non negative integer solutions of a + b + c = 21

Introduction / Context:
This question is a standard example of integer solution counting using the stars and bars method. You are asked to count the number of non negative integer solutions of a linear equation with three variables, which is a staple problem in combinatorics.

Given Data / Assumptions:

• Equation: a + b + c = 21.
• Variables a, b, and c are non negative integers.
• Non negative means each variable can be zero or any positive integer as long as the sum is 21.
• Order of the triple (a, b, c) matters, so different assignments count as different solutions.

Concept / Approach:
The stars and bars method is used to count solutions of x1 + x2 + ... + xk = n where all xi are non negative integers. The formula for the number of solutions is n plus k minus 1 choose k minus 1. Here n is the total sum and k is the number of variables.

Step-by-Step Solution:
Step 1: Identify the total sum n = 21.Step 2: The number of variables is k = 3 corresponding to a, b, and c.Step 3: For non negative integer solutions, the formula is (n + k - 1)C(k - 1).Step 4: Substitute n = 21 and k = 3 into the formula.Step 5: We get (21 + 3 - 1)C(3 - 1) = 23C2.Step 6: Compute 23C2 = 23 * 22 / 2 = 253.Step 7: Thus there are 253 distinct non negative integer solutions for the equation.

Verification / Alternative check:
You can verify the formula by checking a simpler case such as a + b + c = 2. There you could list all solutions and confirm that the stars and bars formula predicts the correct count. Once you trust the formula, applying it to 21 is straightforward.

Why Other Options Are Wrong:

• 123, 321, and 231 are close in size to 253 but none equals the correct binomial coefficient 23C2.
• They likely arise from misusing the formula such as 21C2 or 22C2.

Common Pitfalls:
A major source of error is confusing non negative and positive solutions. For strictly positive solutions, the formula is (n - 1)C(k - 1), which would give a different answer. Another mistake is plugging n and k incorrectly, for example computing (n + k)Ck. Always write the formula clearly and substitute values carefully.

Final Answer:
The total number of non negative integer solutions of a + b + c = 21 is 253, so the correct answer is 253.