# 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. ngockhue

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