Let $S$ be the set of points $(a,b)$ in the coordinate plane, where each of $a$ and $b$ may be $-1$, 0, or 1. How many distinct lines pass t

Let $S$ be the set of points $(a,b)$ in the coordinate plane, where each of $a$ and $b$ may be $-1$, 0, or 1. How many distinct lines pass through at least two members of $S$

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  1. Answer:

    20 Lines

    Step-by-step explanation:

    According to the Question,

    • Given That, Let S be the set of points (a, b) in the coordinate plane, where each of a and b may be -1, 0, or 1.

    Now,  the total pairs of points which can be formed is 9

    And, the line passing through 2 such points 9c2 = 9! / (2! x 7!) = 9×4 ⇒ 36

    Here, We have overcounted all of the lines which pass through three points.

    And, each line that passes through three points will have been counted 3c2 = 3! / 2! ⇒ 3 times

    Now, the sides of the square consist of 3 points. We have counted each side thrice, so 4*2 are repeated.

    • Therefore, the distinct lines pass through at least two members of S is 3 horizontal, 3 vertical, and 2 diagonal lines, so the answer is 36 – 2(3+3+2) = 20 Lines
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