Question

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

• A. $\mathrm{pqr}+(\mathrm{p}-1)(\mathrm{q}-1)(\mathrm{r}-1)$
• B. $\frac{1}{4}\left\{\mathrm{pqr}+(\mathrm{p}-1)(\mathrm{q}-1)(\mathrm{r}-1)\right\}$
• C. $\frac{1}{4}\mathrm{pqr}(\mathrm{p}+1)(\mathrm{q}+1)(\mathrm{r}+1)$
• D. None of these

Answer 1

The correct option is D

None of these

Find the maximum number of parallelograms

Given: $3$ set of parallel lines, containing lines $p,q$ and $r$

The maximum number of parallelograms are $={}^{\mathrm{p}}{\mathrm{C}}_2·{}^{\mathrm{r}}{\mathrm{C}}_2+{}^{\mathrm{p}}{\mathrm{C}}_2·{}^{\mathrm{q}}{\mathrm{C}}_2+{}^{\mathrm{p}}{\mathrm{C}}_2·{}^{\mathrm{r}}{\mathrm{C}}_2$

Hence option (D) is the correct answer.