A parallelogram is cut by two sets of m lines parallel to its sides. The number of parallelograms thus formed, is

Each set is having $(m + 2)$ parallel lines and each parallelogram is formed by choosing two straight lines from the first set and two straight lines from the second set. Two straight lines from the first set can be chosen in ${}^{m+2}C_2$ ways and two straight lines from the second set can be chosen in ${}^{m+2}C_2$ ways. Hence, the total number of parallelograms formed

${=}^{m+2}{C}_2\cdot m+2C_2={\left({ }^{m+2}{C}_2\right)}^2$