How many different four letter words (not necessarily meaningful) can be formed from the distinct letters of the word LOGARITHMS, if repetition of letters is not allowed?

Introduction / Context:
This question checks your understanding of permutations of distinct objects. You are given a word with all different letters and asked to count how many four letter sequences you can build without repeating any letter, which is a straightforward application of permutation principles.

Given Data / Assumptions:

• The word LOGARITHMS has 10 distinct letters: L, O, G, A, R, I, T, H, M, S.
• We must form four letter words using these letters.
• No letter can be repeated within a word.
• Order matters because different arrangements such as GLOT and TLOG are considered different words.

Concept / Approach:
Whenever you are arranging k distinct items selected from n distinct items, and order matters with no repetition, you use permutations. The count is given by nPk = n * (n - 1) * (n - 2) ... up to k factors, or equivalently n! / (n - k)!. Here n = 10 and k = 4.

Step-by-Step Solution:
Number of distinct letters, n = 10.We need the number of 4 letter arrangements: compute 10P4.Use formula: 10P4 = 10 * 9 * 8 * 7.First multiply 10 * 9 = 90, then 90 * 8 = 720, and 720 * 7 = 5040.Therefore, the number of possible four letter words is 5040.

Verification / Alternative check:
Using factorial form: 10P4 = 10! / 6!.10! = 3628800 and 6! = 720.Compute 3628800 / 720 = 5040, which matches the earlier result.

Why Other Options Are Wrong:
2525 and 2052 are not equal to 10P4 and result from incorrect multiplication or division.4521 is another random looking number that does not correspond to any standard combinatorial expression in this setup.

Common Pitfalls:
Using combinations 10C4 instead of permutations, which would ignore the importance of order.Accidentally allowing repetition of letters, which would change the counting logic completely.Forgetting that all letters of LOGARITHMS are distinct and trying to adjust for repeated letters unnecessarily.

Final Answer:
The total number of four letter words is 5040.