How do immunoglobulin genes code for a seemingly infinite variety of antibodies?

Answer: Think of DNA-there are only 4 bases, but different combinations give infinite possibilities. And the same for amino acids and proteins. Immunoglobulins work on a similar principle. Different VDJ segments are combined for the heavy chain of an antibody. Different VJ segments are combined for the light chain. Other ways: Random nucleotides may be added or removed between the gene segments. There are also different combinations of light and heavy chains. Antibodies can be any of the 5 isotypes: IgA, IgM, IgG, IgE, IgD. I'm not going to go into detail as this answer would take too long to write. Do some literature searching on B cells, V (D) J recombination, RAG genes, class switching.