, 2026

The choice of $5$ letters for
each of the $5$ positions within the
string gives a total of $5^5=3125$
different strings in the list.

Of these strings, $\frac15$ begin
with the letter $A$, $\frac15$ begin with the letter $B$, and so on to $E$.

That is, there are $\frac15\times5^5=5^4=625$ strings that
begin with each of the letters $A$
through $E$. Thus, the $3\times625=1875$th string is the last
string that begins with $C$ (this is
$CEEEE$), and the $4\times625=2500$th string is the last
string that begins with $D$ (this is
$DEEEE$).

This tells us that the first letter in the 2026th string is $D$.

Of the $625$ strings that begin
with $D$, $\frac15$ have second letter $A$, $\frac15$ have second letter $B$, and so on to $E$. That is, there are $\frac15\times625=125$ strings having each
of the second letters $A$ through
$E$. Thus, the $1875+125=2000$th string is the last string
that begins with $D$ and has second
letter $A$ (this is $DAEEE$), and the $1875+2\times125=2125$th string is the last
string that begins with $D$ and has
second letter $B$ (this is $DBEEE$).

This tells us that the 2001st string is $DBAAA$, and the first two letters in the
2026th string are $DB$.

Of the $125$ strings that begin
with $DB$, $\frac15$ have third letter $A$, $\frac15$ have third letter $B$, and so on to $E$. That is, there are $\frac15\times125=25$ strings beginning
with $DBA$. The 2001st of these is
$DBAAA$ and the $2000+25=2025$th is $DBAEE$. Thus, the 2026th string is the
next string following $DBAEE$ which
is $DBBAA$, and so the 2026th string
in the list has $2$ Bs.