Question

In how many ways can 8 letters be posted in 5 letter boxes.

• A. \(^{8}P_{5}\)
• B. \(^{8}C_{5}\)
• C. \(8^{5}\)
• D. \(5^{8}\)

Hint:

To avoid confusion between $n^r$ and $r^n$, remember this rule: (Options)\textsuperscript{Items}. Here, the boxes are the options and the letters are the items being moved.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a problem of permutations with repetition allowed. We must determine which set of items has the "freedom of choice." Each letter is a distinct event that must be assigned to a box.

Step 2: Key Formula or Approach:
For $n$ items and $r$ choices for each item, the total number of ways is $r^n$.

Step 3: Detailed Explanation:
1. Consider the first letter: It can be posted in any of the 5 boxes. (5 ways)
2. Consider the second letter: It can also be posted in any of the 5 boxes. (5 ways)
3. This continues for all 8 letters.
4. Total ways = \( 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 5^8 \).

Step 4: Final Answer:
The total number of ways to post the letters is \(5^{8}\).