To solve this problem, we need to list all the combinations of five objects $x, y, z, s, t$ taken two at a time.

Explanation:

When we are choosing 2 objects from 5 distinct objects, the combination formula to find the total number of combinations is:

$\binom{5}{2} = \frac{5!}{2! \cdot (5-2)!} = \frac{5 \times 4}{2 \times 1} = 10$

This tells us there are 10 possible combinations, and these combinations are all the different ways we can pick 2 objects from the 5 available.

Finding the Correct Answer:

We need to choose the correct answer from the options provided:

• Option A: Lists all possible pairs including repeated letters, like $xx, yy, zz$, etc. This is not correct since we are only interested in combinations of distinct objects.

• Option B: Lists all possible pairs with distinct letters, but also includes each pair in both orders (like $xy$ and $yx$), which is a permutation and not a combination. Therefore, this option is incorrect.

• Option C: Does not provide combinations but simply lists the original objects. This is not what we want.

• Option D: Lists all the unique pairs of distinct objects in alphabetical order, which is the correct set of combinations.

Correct Answer:

Option D: $\text{xy, xz, xs, xt, yz, ys, yt, zs, zt, st}$

Verification:

The answer has all the combinations of the objects $x, y, z, s, t$ taken two at a time, without repetition and considering order does not matter.

Conclusion:

• The correct answer is Option D.

Would you like to know more details or have any questions?

Additional Questions:

1. What is the difference between combinations and permutations?
2. How do we calculate combinations for a different number of objects?
3. How does the formula for combinations change when considering objects with repetitions?
4. Can you explain how to find combinations using a binomial coefficient?
5. How would the answer change if we were selecting three objects at a time?

Tip: Remember that in combinations, the order does not matter, while in permutations, the order is significant.