In this question, we are asked to identify the family of plants where the stamens are fused with each other by their anthers and also with the petals. The correct answer is option 3 - Compositae.

Compositae, also known as Asteraceae, is a family of flowering plants that includes species such as daisies, sunflowers, and dandelions. In this family, the stamens are fused with each other by their anthers, forming a ring or tube structure. Additionally, they are also fused with the petals, creating a unique characteristic of this family.

Leguminosae (option 1) is the family of plants that includes species such as peas and beans, but the stamens in this family are not fused with each other or with the petals. Liliaceae (option 2) is the family of plants that includes lilies, but again, the stamens are not fused in this family. Euphorbiaceae (option 4) is the family of plants that includes species like the poinsettia, but the stamens are not fused in this family either.

Therefore, the correct answer is option 3 - Compositae.