Geometry & Mensuration
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Q.

\(S_1\)and \(S_2\) are two  sets of parallel lines.  The number of lines in \(S_1\) is greater than the number of \(S_2\) They intersect at 12 points.  The number of parallelograms that \(S_1\) and \(S_2\)  may form is :

 A.

12 or 6

 B.

8 or 4

 C.

18

 D.

18 or 15

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Solution:
Option(D) is correct

Let there be \(m\) lines is \(S_1\) and \(n\) lines in \(S_2\).The lines will intersect in \(mn\) points.

Therefore \(mn = 12\).

Case (1), \(m = 6, n = 2\) (or) \(m = 3, n = 4,\)

The number of parallelograms that can be formed is

\(=^6C_2\times ^2C_2\)

\(=15\times 1\)

\(=15\)
Case (2), \( m = 4, n = 3,\)

The number of parallelograms that can be formed is

\(=^4C_2\times ^3C_2\)

\(=6\times 3\)

\(=18\)


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