Question

Suppose a triangle is formed by $x + y = 10$ and the coordinate axes. Then the number of points $(x, y)$ where $x$ and $y$ are natural numbers, lying inside the triangle is

• A. 36
• B. 55
• C. 45
• D. 30

Hint:

For counting integer solutions inside a triangle bounded by axes and a line, use inequality and summation logic.

Answer 1

The Correct Option is A

We want the number of natural number solutions to $x + y<10$.
For each $x = 1$ to $8$, $y$ can range from $1$ to $9 - x$:
$x = 1 \Rightarrow y = 1$ to $8$ → 8 values
$x = 2 \Rightarrow y = 1$ to $7$ → 7 values
...
$x = 8 \Rightarrow y = 1$ → 1 value
Total = $8 + 7 + \cdots + 1 = \dfrac{8 \cdot 9}{2} = 36$