Final answer:
There are 24 distinct ways to arrange 5 keys on a ring when accounting for the ring's circular symmetry and the rotations of the keys are considered the same arrangement.
Step-by-step explanation:
The question asks, "In how many ways can 5 keys be put on a ring?" This problem falls under the category of combinatorics, a branch of mathematics focused on counting and arranging objects. Since a key ring is circular, the arrangement of keys is unaffected by rotations. Thus, the problem is analogous to arranging the keys in a circle.
Considering the rotations as identical, we can fix one key and arrange the remaining four keys around it. This reduces our problem to ordering four unique items, which is a permutation problem. The number of permutations of 'n' unique items is 'n!'. Therefore, for 4 keys, the number of ways they can be arranged is 4!.
Applying this formula, we calculate the number of arrangements as 4! = 4 × 3 × 2 × 1 = 24.
So, there are 24 distinct ways to arrange 5 keys on a ring, considering the ring's circular nature and rotational symmetry.