Final answer:
The equation x₁+x₂+x₃=11 has 78 non-negative integer solutions.
Step-by-step explanation:
The equation x₁+x₂+x₃=11 represents the number of ways we can distribute 11 identical objects into 3 distinct groups. To solve this problem, we can use the concept of stars and bars. Imagine representing each object as a star and the groups as bars. We can place the bars in between the stars to divide them into different groups.
In this case, we have 11 stars and 2 bars, which results in 13 total objects. The number of solutions is then the number of ways we can choose the positions for the 2 bars among the 13 objects, which can be calculated using combinations.
Using the formula for combinations, we can calculate the number of solutions:
C(13, 2) = 13! / (2! * (13-2)!) = (13 * 12) / 2 = 78
Therefore, the equation x₁+x₂+x₃=11 has 78 non-negative integer solutions.