Step 1: Understand digestion of circular plasmid DNA.
For a circular plasmid, if there are \(n\) restriction sites for an enzyme, complete digestion gives \(n\) DNA fragments.
Step 2: Identify the original BamHI sites.
Initially, the plasmid has four BamHI sites:
\[
p,\ q,\ r,\ s
\]
Step 3: Apply the mutation condition.
The mutation causes loss of BamHI site \(r\).
So, the remaining BamHI sites are
\[
p,\ q,\ s
\]
Step 4: Determine the number of physical fragments.
Since three BamHI sites remain on the circular plasmid, complete digestion produces three physical fragments.
Step 5: Find the fragment sizes.
From the figure:
\[
p\to q=2.4\ \text{kb}
\]
\[
q\to r=1.2\ \text{kb}, \quad r\to s=2.2\ \text{kb}
\]
Since \(r\) is lost, the fragment from \(q\) to \(s\) becomes
\[
1.2+2.2=3.4\ \text{kb}
\]
Also,
\[
s\to p=3.4\ \text{kb}
\]
Step 6: Identify bands seen on agarose gel.
The fragments are
\[
2.4\ \text{kb},\quad 3.4\ \text{kb},\quad 3.4\ \text{kb}
\]
The two \(3.4\ \text{kb}\) fragments migrate together and appear as a single band.
Step 7: Final conclusion.
Thus, the number of distinct DNA fragments seen on agarose gel is
\[
\boxed{2}
\]