How many binary operations can be defined on a set {1, 2, 3, 4}?
(A) 4³
(B) 4⁴
(C) 4¹⁶
(D) 4²

To solve this question, let's first understand the concept of a binary operation on a set.

A **binary operation** on a set is a rule that combines two elements from the set to produce a third element, also from the same set. For example, if we have a set \( S = \{1, 2, 3, 4\} \), a binary operation will take two elements from this set and map them to a result that is also from the set.

### Step-by-step breakdown:

1. **Number of elements in the set**: The set \( S = \{1, 2, 3, 4\} \) has 4 elements.

2. **Choosing two elements for the operation**: A binary operation takes two elements from the set. For each pair of elements, the result of the operation is also an element from the set.

For each pair of elements in the set, there are 4 possible outcomes (since the result must be an element of the set, and the set has 4 elements).

3. **Total number of pairs**: There are \( 4 \times 4 = 16 \) possible pairs of elements from the set (since we are choosing two elements from 4, and the order of the elements matters in a binary operation).

4. **Total number of binary operations**: For each of the 16 possible pairs of elements, there are 4 possible results. Therefore, the total number of binary operations is:
\[
4^{16}
\]

### Answer:

The correct answer is **(C) \( 4^{16} \)**.