This problem involves the use of three restriction enzymes ('a', 'b', and 'c') on a circular plasmid. Let’s break down the sequence of reactions:
1. Enzymes 'a' and 'b':
When enzymes 'a' and 'b' are used together, they cleave the plasmid at their respective unique sites, resulting in two fragments of equal size. Since the plasmid is circular, this creates two linear fragments of equal size.
2. Enzyme 'c':
Enzyme 'c' cleaves only one of the fragments produced by the combination of enzymes 'a' and 'b'. Since the fragments from 'a' and 'b' are of equal size, enzyme 'c' will cleave one of these fragments into two equal fragments.
Thus, after enzyme 'c' acts, we now have three distinct fragments:
- Two from enzyme 'a' and 'b' acting together (of equal size),
- One more generated by enzyme 'c', cleaving one of the previous fragments.
3. Analysis by Agarose Gel Electrophoresis:
In agarose gel electrophoresis, each fragment will appear as a separate band. Therefore, with three fragments in total, there will be 3 bands observed in the gel.
Thus, the number of bands observed in the gel is:
\[
\boxed{3}
\]