Question:
If \(m\) parallel lines in a plane are intersected by a family of \(n\) parallel lines, then the number of parallelograms that can be formed is
If \(m\) parallel lines in a plane are intersected by a family of \(n\) parallel lines, then the number of parallelograms that can be formed is
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A parallelogram is formed by choosing 2 lines from each parallel set.
Updated On: Apr 23, 2026
- \(\frac{1}{4} mn(m-1)(n-1)\)
- \(\frac{1}{2} mn(m-1)(n-1)\)
- \(\frac{1}{4} m^2 n^2\)
- None of these
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The Correct Option is A
Solution and Explanation
Step 1: Formula / Definition}
\[ \text{Number of parallelograms} = \binom{m}{2} \times \binom{n}{2} \]
Step 2: Calculation / Simplification}
\[ \binom{m}{2} = \frac{m(m-1)}{2} \]
\[ \binom{n}{2} = \frac{n(n-1)}{2} \]
\[ \text{Total} = \frac{m(m-1)}{2} \times \frac{n(n-1)}{2} = \frac{1}{4} mn(m-1)(n-1) \]
Step 3: Final Answer
\[ \frac{1}{4} mn(m-1)(n-1) \]
\[ \text{Number of parallelograms} = \binom{m}{2} \times \binom{n}{2} \]
Step 2: Calculation / Simplification}
\[ \binom{m}{2} = \frac{m(m-1)}{2} \]
\[ \binom{n}{2} = \frac{n(n-1)}{2} \]
\[ \text{Total} = \frac{m(m-1)}{2} \times \frac{n(n-1)}{2} = \frac{1}{4} mn(m-1)(n-1) \]
Step 3: Final Answer
\[ \frac{1}{4} mn(m-1)(n-1) \]
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