Key Concepts and Formulas

• Lattice Points: Points with integer coordinates are called lattice points or integral points.
• Inequalities for Regions: The interior of a region bounded by lines can be described using strict inequalities ($<$ or $>$).
• Sum of First n Natural Numbers: The sum of the first $n$ natural numbers is given by the formula: $S_n = \frac{n(n+1)}{2}$

Step-by-Step Solution

Step 1: Define the Triangle and its Boundaries

We are given a triangle with vertices at $(0, 0)$, $(0, 41)$, and $(41, 0)$. We need to find the number of lattice points in the interior of this triangle. The sides of the triangle are defined by the following lines:

• $x = 0$ (y-axis)
• $y = 0$ (x-axis)
• $x + y = 41$ (line connecting $(0, 41)$ and $(41, 0)$)

Step 2: Define the Inequalities for the Interior of the Triangle

For a point $(x, y)$ to lie in the interior of the triangle, it must satisfy the following strict inequalities:

• $x > 0$ (to the right of the y-axis)
• $y > 0$ (above the x-axis)
• $x + y < 41$ (below the line $x + y = 41$)

Since we are looking for integer coordinates, $x$ and $y$ must be integers.

Step 3: Determine the Range of Possible x Values

Since $x$ must be a positive integer, the smallest possible value for $x$ is $1$. We also have the constraint $x + y < 41$. Since $y > 0$, the smallest possible integer value for $y$ is $1$. Therefore, $x + 1 < 41$, which means $x < 40$. Since x must be an integer, $x \le 39$. Thus, the possible integer values for $x$ are $1, 2, 3, ..., 39$.

Step 4: Determine the Range of Possible y Values for Each x

For each value of $x$, we need to find the possible integer values of $y$ that satisfy the inequalities. We know $y > 0$ and $x + y < 41$, which means $y < 41 - x$. Combining these, we have $0 < y < 41 - x$. Since y must be an integer, we have $1 \le y \le 40 - x$.

Step 5: Count the Number of Possible y Values for Each x

For each integer value of $x$ from $1$ to $39$, the number of possible integer values for $y$ is $(40 - x) - 1 + 1 = 40 - x$.

• If $x = 1$, then $1 \le y \le 39$, so there are $39$ possible values for $y$.
• If $x = 2$, then $1 \le y \le 38$, so there are $38$ possible values for $y$.
• If $x = 3$, then $1 \le y \le 37$, so there are $37$ possible values for $y$.
• ...
• If $x = 39$, then $1 \le y \le 1$, so there is $1$ possible value for $y$.

Step 6: Calculate the Total Number of Lattice Points

The total number of lattice points in the interior of the triangle is the sum of the number of possible $y$ values for each $x$:

$\text{Total points} = 39 + 38 + 37 + \dots + 1$

This is the sum of the first 39 natural numbers, which can be calculated using the formula:

$\text{Total points} = \frac{39(39 + 1)}{2} = \frac{39 \times 40}{2} = 39 \times 20 = 780$

Common Mistakes & Tips

• Strict vs. Non-Strict Inequalities: Be careful to use strict inequalities ($<$ or $>$) for "interior" points and non-strict inequalities ($\le$ or $\ge$) for points "on or within" the region.
• Counting Integers in a Range: When counting the number of integers in a range $a \le k \le b$, remember to include both endpoints by using the formula $b - a + 1$.
• Formula Application: Recognize common patterns like the sum of the first $n$ natural numbers to simplify calculations.

Summary

We found the number of lattice points in the interior of the triangle with vertices $(0, 0)$, $(0, 41)$, and $(41, 0)$ by defining the region using inequalities, iterating through possible integer values for $x$, and calculating the range of possible integer values for $y$ for each $x$. Finally, we summed the number of $y$ values for each $x$ to find the total number of lattice points, which is 780.

Final Answer

The final answer is \boxed{780}, which corresponds to option (B).