Five distinct letters are to be posted into four distinct letter boxes. If each letter can go into any box independently of the others, in how many different ways can all five letters be posted?

Introduction / Context:
This problem is about distributing distinct objects into distinct boxes when there is no restriction on how many objects each box can receive. Each of the five letters can be posted into any of the four letter boxes independently.

Given Data / Assumptions:

• Number of letters = 5.
• Number of letter boxes = 4.
• Each letter can go into any of the 4 boxes.
• More than one letter can go into the same box.
• Letters and boxes are all distinct.

Concept / Approach:
For each letter, there are 4 possible choices of box. Since the placement of each letter is independent of the others, we can apply the multiplication rule of counting, raising the number of choices to a power equal to the number of letters.

Step-by-Step Solution:
Step 1: Focus on one letter at a time. For letter 1, there are 4 possible boxes. Step 2: For letter 2, again there are 4 possible boxes, regardless of where letter 1 went. Step 3: The same is true for each of the remaining letters; each has 4 choices of box. Step 4: Since these choices are independent, total number of ways = 4 * 4 * 4 * 4 * 4 = 4^5. Step 5: Compute 4^5 = 4 * 4 * 4 * 4 * 4 = 1024.

Verification / Alternative check:
You can view each distribution as a 5 digit code in base 4 where each position represents a letter and each digit represents the chosen box. The total number of such codes is 4^5, which again equals 1024.

Why Other Options Are Wrong:
512 equals 2^9 or 8^3 and has no direct connection to the structure 4^5 in this context. 625 is 5^4, which corresponds to the reversed base. The value 20 would correspond to combinations or some limited case that does not fit this problem.

Common Pitfalls:
The main error is to treat letters as identical or to think in terms of combinations rather than independent placements. Some learners also mistakenly divide by factorials, which is not appropriate because different assignments of letters to boxes are distinct arrangements.

Final Answer:
There are 1024 different ways to post the five letters into the four letter boxes.