Ten distinct letters are to be posted into five different post boxes. Each post box can receive any number of letters (each can hold more than ten letters without any restriction). In how many different ways can all the ten letters be posted into these five distinct post boxes?

Introduction / Context:
This problem tests a very common idea from permutations and combinations, namely distributing distinct objects (letters) into distinct groups (post boxes) when each group can hold any number of objects. Such questions appear frequently in aptitude exams to check whether the student can quickly recognise the correct counting model and avoid overcomplicating the scenario with unnecessary conditions.

Given Data / Assumptions:

• There are 10 distinct letters that need to be posted.
• There are 5 distinct post boxes available.
• Each letter must go into exactly one post box.
• Each post box can hold any number of letters; there is no upper limit.

Concept / Approach:
The key concept is to observe that each letter has 5 independent choices because it can be posted into any of the 5 post boxes. When choices for different letters are independent, the total number of ways is obtained by multiplying the number of choices for each letter. Thus, the basic rule used here is the fundamental principle of counting: if one action can be done in m ways and another in n ways independently, then together they can be done in m * n ways.

Step-by-Step Solution:
Step 1: Consider the first letter. It can be placed in any one of the 5 post boxes, so it has 5 choices.Step 2: The second letter is also free to go into any one of the 5 post boxes, again giving 5 choices, independent of the first letter.Step 3: Similarly, each of the remaining 8 letters has 5 choices, because there is no capacity restriction.Step 4: By the fundamental principle of counting, the total number of ways is 5 * 5 * 5 * ... (10 times) which is 5^10.Step 5: Therefore, the required number of ways to post 10 letters into 5 post boxes is 5^10.
Verification / Alternative check:
An alternative way to think of the same result is to treat each letter as a position in a 10 length string, where for each position we choose one of the 5 boxes as its destination. This again clearly gives 5 choices for each of the 10 positions, hence 5^10. This matches our earlier calculation, so the answer is consistent.

Why Other Options Are Wrong:
10^5 is incorrect because it would correspond to distributing 5 letters into 10 boxes, which is not the given situation here. 5P5 is just 5! which counts permutations of the boxes, not distributions of letters. 5C5 counts the number of ways to choose 5 boxes out of 5, which is 1 and is unrelated to the actual question. "None of these" is not correct because one of the listed expressions, namely 5^10, is correct.

Common Pitfalls:
Students sometimes confuse this model with permutations like 10! or combinations like 10C5, both of which would be wrong because we are not arranging letters in order; we are allocating letters into boxes. Another pitfall is to assume there must be a maximum of one letter or some fixed number of letters per box, which is explicitly not the case here. Correctly recognising that each letter independently chooses a box is essential.

Final Answer:
The total number of ways to post the 10 letters into the 5 post boxes is 5^10.