A human muscle cell has the same DNA inside its nucleus as a human nerve cell, even though these cells are very different in structure and function. How is this possible?

This is because in every cell, certain genes are expressed more than others. For instance, in a cell making up a human muscle cell, the cell would only express the genes appropriate to muscle function and not express the genes appropriate to that of a nerve cell. The opposite would be true for a nerve cell, likewise.