The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of four parallel lines, is

Answer is (d). In the diagram, let’s count the parallelograms one by one. Case I - Parallelograms of 1 × 1 (ABFE type) - ABFE, BCGF, CDHG, EFJI, FGKJ, GHLK, IJNM, JKON, KLPO – total 9. Case II - Parallelograms of 1 × 2 (ACGE type) - ACGE, BDFH, EGKI, FHLJ, IKOM, JLPN – total 6. Case III - Parallelogram of 2 × 1 (ABJI type) - ABJI, EFNM, BCKJ, FGON, CDLK, GHPO – total 6. Case IV - Parallelograms of 1 × 3 (ADHE type) - ADHE, EHLI, ILPM – total 3. Case V - Parallelograms of 3 × 1 (ABNM type) - ABNM, BCON, CDPO – total 3. Case VI - Parallelograms of 2 × 2 (ACKI type) - ACKI, BDLJ, EGOM, FHPN – total 4. Case VII - Parallelograms of 3 × 2 (ADLI type) - ADLI, EHPM – total 2. Case VIII - Parallelograms of 2 × 3 (ACOM type) - ACOM, BDPN – total 2. Case IX -Parallelograms of 3 × 3 (ADPM type) – ADPM – total 1. Total 36. A much shorter method is by using permutations and combinations. Select any two of the first set of 4 lines. That can be done in 4C2 ways. Now select any two of the second set of 4 lines. That can also be done in 4C2 ways. So the total number of ways of doing it = 4C2 x 4C2 = 6 x 6 = 36 ways.