The word "PROBLEM" has 7 distinct letters.\nIn how many different ways can all the letters be rearranged to form 7 letter words if no letter is repeated?

Introduction / Context:
This is a pure permutation question involving a word with all distinct letters. We must count the number of ways to rearrange all letters of the word, which directly applies the factorial rule for permutations when there are no repetitions.

Given Data / Assumptions:

• Word: PROBLEM.
• Letters: P, R, O, B, L, E, M, all distinct.
• Total letters = 7.
• All 7 letters must be used in each arrangement.
• No letter is repeated within an arrangement.
• Different orders of the same letters are considered different words.

Concept / Approach:
For n distinct objects arranged in a row with all used exactly once, the number of permutations is simply n!. No adjustment for repetition is needed here because every letter appears exactly once. Hence we compute 7! for this problem.

Step-by-Step Solution:
Total letters to arrange = 7. Number of permutations of 7 distinct letters = 7!. Compute 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1. First, 7 * 6 = 42 and 5 * 4 = 20. Then 42 * 20 = 840. Next multiply by 3 * 2 * 1 = 6. Finally, 840 * 6 = 5040.

Verification / Alternative check:
We can think position by position. There are 7 choices for the first position, 6 for the second, 5 for the third and so on until there is only 1 choice left for the last position. The total number is therefore 7 * 6 * 5 * 4 * 3 * 2 * 1, which again equals 7!. This repeated reasoning confirms that 5040 is the correct count of different 7 letter arrangements.

Why Other Options Are Wrong:

• 49: This is 7^2 and could come from misinterpreting the structure of the word or using an incorrect formula.
• 823543: This equals 7^7 and would be the count if each of the 7 positions could independently be any of 7 letters with repetition allowed, which is not the case.
• 343: This equals 7^3 and has no connection to the correct permutation logic here.

Common Pitfalls:
The main error learners make is mixing up permutations with combinations or using exponent rules like 7^n without thinking about whether repetition is allowed. Another issue is forgetting that order matters when forming words or sequences. When every letter is distinct and all must be used once, the factorial rule n! is a simple and reliable tool.

Final Answer:
The number of different 7 letter words that can be formed from "PROBLEM" is 5040.