Analysis:
To determine which numbers can be the number of elements in a finite Boolean algebra, we need to understand the properties and characteristics of Boolean algebras.

Definition of a Boolean Algebra:
A Boolean algebra is a mathematical structure consisting of a set of elements, a set of binary operations (usually denoted as ∧, ∨, and ¬), and a set of axioms or rules that govern the behavior of these operations. In a Boolean algebra, the operations ∧ (meet or intersection), ∨ (join or union), and ¬ (complement or negation) must satisfy certain properties.

Properties of a Boolean Algebra:
1. Closure under meet and join: For any two elements x and y in the Boolean algebra, the meet x ∧ y and join x ∨ y must also be elements of the algebra.
2. Associativity: The meet and join operations are associative, i.e., (x ∧ y) ∧ z = x ∧ (y ∧ z) and (x ∨ y) ∨ z = x ∨ (y ∨ z) for any elements x, y, and z.
3. Commutativity: The meet and join operations are commutative, i.e., x ∧ y = y ∧ x and x ∨ y = y ∨ x for any elements x and y.
4. Distributivity: The meet and join operations distribute over each other, i.e., x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) and x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) for any elements x, y, and z.
5. Identity elements: There exist two elements 0 and 1 in the Boolean algebra such that for any element x, x ∧ 0 = 0 and x ∨ 1 = 1.
6. Complement: For every element x in the Boolean algebra, there exists another element ¬x (complement of x) such that x ∧ ¬x = 0 and x ∨ ¬x = 1.

Number of Elements in a Finite Boolean Algebra:
The number of elements in a finite Boolean algebra can be determined by considering the number of elements in each possible subset of the set of elements. Let n be the number of elements in the set. Since each element can either be included or excluded from a subset, the total number of possible subsets is 2^n.

Answer:
The number of elements in a finite Boolean algebra can be any non-negative power of 2. Therefore, the correct answer is option A) 2^n.