Question

There are 3 set of parallel lines containing respectively $p$ lines, $q$ lines and $r$ lines respectively. The greatest number of parallelograms that can be formed by the system

• A. $p q r+(p-1)(q-1)(r-1)$
• B. $\frac{1}{4}\{p q r+(p-1)(q-1)(r-1)\}$
• C. $\frac{1}{4} \operatorname{pqr}(p+1)(q+1)(r+1)$
• D. none of these

Answer 1

Answer: (D)

Number of parallelograms
$=\left({ }^p C_2\right)\left({ }^q C_2\right)+\left({ }^q C_2\right)\left({ }^r C_2\right)+\left({ }^r C_2\right)\left({ }^p C_2\right)$