List all the combinations of five objects a, b, c, d, and e taken two at a time. What is ${ }_{5} \mathrm{C}_{2}$ ?

List all the combinations of five objects a, b, c, d, and e taken two at a time. Choose the correct answer below.
A. ab, ac, ad, ae, bc, bd, be, cd, ce, de
B. aa, ab, ac, ad, ae, bb, bc, bd, be, cc, cd, ce, dd, de, ee
C. $a, b, c, d, e$
D. $a b, a c$, ad, ae, ba, bc, bd, be, ca, cb, cd, ce, da, db, dc, de,

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 \(\boxed{10}\).