Let's analyze this step by step. We need to find all points (x,y) where both x and y are integers from 0 to 10, and their sum x+y is also between 0 and 10. The key constraint is x+y ≤ 10, which creates a triangular region in our coordinate system.
Now let's count systematically. For each fixed value of x, we determine how many valid y values exist. When x equals 0, y can range from 0 to 10, giving us 11 points. When x equals 1, y can range from 0 to 9, giving us 10 points. This pattern continues until x equals 10, where y can only be 0.
We need to sum the arithmetic sequence: 11 plus 10 plus 9 and so on down to 1. This is the sum of the first 11 positive integers. Using the formula for the sum of consecutive integers, we get 11 times 12 divided by 2, which equals 66.
Let's verify our solution. We can check specific cases: when x equals 0, y can be any value from 0 to 10, giving us 11 points. When x equals 5, y can range from 0 to 5, giving us 6 points. When x equals 10, y can only be 0, giving us 1 point. This confirms our pattern and our final answer of 66 points.
To summarize our solution: We identified the constraints as x and y both ranging from 0 to 10, with the additional constraint that x plus y must not exceed 10. We counted systematically, finding that for each value of x, there are 11 minus x valid y values. This gives us the sum 11 plus 10 plus 9 down to 1, which equals 66. Therefore, the answer to this AMC problem is 66.