Step 1: Rule for a circular plasmid.
Cutting a circle at $n$ sites gives $n$ pieces. We start with four BamHI sites $p, q, r, s$.
Step 2: Apply the mutation.
Losing site $r$ leaves three sites, $p$, $q$ and $s$, so digestion now makes three physical pieces.
Step 3: Use the fragment sizes.
Removing $r$ merges its two neighbouring pieces, $1.2$ kb and $2.2$ kb, into one $3.4$ kb piece. The piece from $s$ round to $p$ is also $3.4$ kb, and the $p$ to $q$ piece is $2.4$ kb.
Step 4: Count the bands on the gel.
The two $3.4$ kb pieces are the same length, so they travel together and show as one band. Together with the $2.4$ kb band that gives two visible bands.
Step 5: Answer.
\[ \boxed{2} \]