Let 𝐿1 be the language represented by the regular expression 𝑏∗𝑎𝑏∗…

Key insight: the language L1 consists exactly of all strings over {a,b} that contain an odd number of a's (at least one).

Reason: The expression b* a b* (a b* a b*)* has one initial a and each repetition adds two a's, so the total number of a's is 1 + 2k (always odd).

• Length 0: 0 strings in L1 (no a's).

• Length 1: 1 string with an odd number of a's (a).

• Length 2: 2 strings with an odd number of a's (exactly one a).

• Length 3: 4 strings with an odd number of a's.

• Length 4: 8 strings with an odd number of a's.

Total: 1 + 2 + 4 + 8 = 15 strings.