Step 1 :The question is asking for all combinations of five objects a, b, c, d, and e taken two at a time. In combinations, order does not matter, so 'ab' and 'ba' would be considered the same combination.
Step 2 :The combinations of the five objects taken two at a time are: ('a', 'b'), ('a', 'c'), ('a', 'd'), ('a', 'e'), ('b', 'c'), ('b', 'd'), ('b', 'e'), ('c', 'd'), ('c', 'e'), ('d', 'e'). This matches with option A in the question.
Step 3 :Now, let's calculate ${ }_{5} \mathrm{C}_{2}$ which is the number of ways to choose 2 objects from 5.
Step 4 :The number of such combinations, ${ }_{5} \mathrm{C}_{2}$, is 10.
Step 5 :Final Answer: The combinations of five objects a, b, c, d, and e taken two at a time are ab, ac, ad, ae, bc, bd, be, cd, ce, de. The number of such combinations, ${ }_{5} \mathrm{C}_{2}$, is \(\boxed{10}\).