How can an antibiotic resistant strain of bacteria occur?

Because of crossing over in mitosis and the exchange of genetic material between bacterial cells there a millions of variations between the bacterial cells.

The two or three cells that carry the combination of genes that grant them resistance for the antibody survive and pass their successful gene to their descendants. This is selection from existing variations.

Investigators found that in strains of bacteria had developed resistance by dropping from its chromosomes a gene called kat G. The cells had paid a price to defend themselves from the antibiotic . The cells had made an evolutionary trade-off giving up part of their own adaptive equipment for the sake of survival (The Beak of the Finch Jonathan Weiner 1994 pages 258-260)

Research indicates that bacteria develop resistance by mutations that result from the loss of genetic information or the selection of existing combinations of genetic information.

Antibiotic-resistant strains of bacteria can occur through several mechanisms, including spontaneous mutations that confer resistance, horizontal gene transfer (such as through plasmids or transposons), and selective pressure from the overuse or misuse of antibiotics, which allows resistant bacteria to survive and proliferate.