2. Let $T \subseteq\{1,2, \cdots, 25\}$. If for any two distinct elements $a, b (a \neq b)$ in $T$, their product $ab$ is not a perfect square, find the maximum number of elements in $T$, and the number of subsets $T$ that satisfy this condition.
Prompt: Divide the set $\{1,2, \cdots, 25\}$ into several subsets such that the product of any two elements in the same subset is a perfect square, and the product of any two elements in different subsets is not a perfect square.
To maximize the number of elements in $T$, let
$$
\begin{array}{l}
A_{1}=\{1,4,9,16,25\}, A_{2}=\{2,8,18\}, \\
A_{3}=\{3,12\}, A_{4}=\{5,20\}, A_{5}=\{6,24\}, \\
A_{6}=\{7\}, A_{7}=\{10\}, A_{8}=\{11\}, A_{9}=\{13\}, \\
A_{10}=\{14\}, A_{11}=\{15\}, A_{12}=\{17\}, A_{13}=\{19\}, \\
A_{14}=\{21\}, A_{15}=\{22\}, A_{16}=\{23\} .
\end{array}
$$

Taking one number from each of these 16 sets to form the set $T$ meets the requirement.
Therefore, the set $T$ can have at most 16 elements.
If the set $T$ has more than 16 elements, then there must be two elements from the same set $A_{k}(k=1,2, \cdots, 16)$, thus, the product of these two elements will be a perfect square.
Hence, the number of sets $T$ that satisfy the requirement is
$$
5 \times 3 \times 2 \times 2 \times 2=120 \text { (sets). }
$$