GATE CS 2024 Set 2 Q.62: Regular Expressions, Finite Automata and Regular Languages

Let $L_1$ be the language represented by the regular expression $b^*ab^*(ab^*ab^*)^*$ and $L_2 = \{w \in (a + b)^* \mid |w| \le 4\}$ , where $|w|$ denotes the length of string $w$ . The number of strings in $L_2$ which are also in $L_1$ is ___________

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Official answer and solution

Official answer: 15
From: Official GATE 2024 answer key

Solution

Observe that $L_1 = b^*ab^*(ab^*ab^*)^*$ is the set of all strings over $\{a, b\}$ with an odd number of $a$ 's, and count such strings of length $0 \le |w| \le 4$ .

The regular expression $L_1 = b^*ab^*(ab^*ab^*)^*$ matches all strings over $\{a, b\}$ that contain an odd number of $a$ 's ( $1, 3, 5, \dots$ occurrences of $a$ ).

For each string length $k \in \{0, 1, 2, 3, 4\}$ in $L_2$ :

$k = 0$ : $\epsilon$ has $0$ $a$ 's ( $0$ strings).
$k = 1$ : $2^{1-1} = 1$ string (a).
$k = 2$ : $2^{2-1} = 2$ strings (ab, ba).
$k = 3$ : $2^{3-1} = 4$ strings ( $\binom{3}{1} + \binom{3}{3} = 3 + 1 = 4$ ).
$k = 4$ : $2^{4-1} = 8$ strings ( $\binom{4}{1} + \binom{4}{3} = 4 + 4 = 8$ ).

Total number of strings in $L_1 \cap L_2 = 0 + 1 + 2 + 4 + 8 = 15$ .

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2024 Computer Science and Information Technology - Set 2 (CS2)
Question: 62 of 65
Section: Computer Science and Information Technology

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