How many points with integer coordinates are in the region \(x\geq 1\), \(y\geq 1\), and \(x+y\leq 5\)?

Correct answer: 10

Since \(x\geq 1\) and \(x+y\leq 5\), the possible integer values of \(x\) are \(1,2,3,4\). For these values, the allowable positive integer values of \(y\) are respectively \(1\) to \(4\), \(1\) to \(3\), \(1\) to \(2\), and \(1\) to \(1\). Therefore, the number of points is \(4+3+2+1=10\). An option such as \(8\) misses some valid boundary points. Exam tip: when an inequality includes equality, count the points on the boundary line as well.