Final answer:
A deterministic finite automaton (DFA) with a finite number of states cannot be designed to accept strings with an equal number of zeroes and ones if the length is unrestricted, as it would require infinite states to track the count. For limited length strings, a DFA could be designed but it's complex. The appropriate automaton for this unrestricted language is the Pushdown Automaton (PDA).
Step-by-step explanation:
The question asks to design a deterministic finite automaton (DFA) that accepts a string containing an equal number of zeroes and ones over the alphabet \{0, 1\}. The correct answer to the student's question is: d) The DFA accepts strings with an equal number of zeroes and ones.
Designing such a DFA is quite complex because it needs to keep track of the number of zeroes and ones that have been input and these could be presented in any order and any length. This tracking would typically require an infinite number of states, which is not possible in a DFA since they have a finite number of states. However, for an educational purpose, if we limit the length of the input string (for example, strings of length 2), then we could design a DFA for this limited scenario with four states, including the start state, accepting state, and two intermediate states to track the number of zeroes and ones.
Unfortunately, for an unrestricted length, a DFA cannot be designed for this language because the DFA does not have memory and cannot count. The correct type of automaton for this type of language with an unrestricted length is a Pushdown Automaton (PDA), which has memory in the form of a stack and can therefore keep track of an arbitrary number of symbols.