How many different 4 letter words, with or without meaning, can be formed from the letters of the word LOGARITHMS if no letter is repeated beyond what is available?

Introduction / Context:
This question checks your ability to form permutations of a fixed length from a larger set of distinct letters. The word LOGARITHMS consists of 10 different letters. We are asked to form 4 letter sequences (words) using these letters without repetition. It is a direct application of permutations of distinct objects taken r at a time, written as nPr.

Given Data / Assumptions:
Word: LOGARITHMS. Letters: L, O, G, A, R, I, T, H, M, S. Total distinct letters available = 10. We need to form 4 letter words. No letter may be repeated within a single word. Order of letters within the word matters because different orders give different words.

Concept / Approach:
We use permutations because the arrangement of selected letters matters. We have 10 distinct letters and we are forming ordered arrangements of length 4. The number of such permutations is given by 10P4, which equals 10 * 9 * 8 * 7. This formula arises from counting choices step by step: first position has 10 choices, second has 9 remaining choices, and so on.

Step-by-Step Solution:
Step 1: Compute 10P4 for permutations of 10 distinct letters taken 4 at a time. Step 2: 10P4 = 10 * 9 * 8 * 7. Step 3: Calculate 10 * 9 = 90. Step 4: Then 90 * 8 = 720. Step 5: Finally, 720 * 7 = 5040. Step 6: Therefore, the number of different 4 letter words that can be formed is 5040.

Verification / Alternative check:
We can also check using the factorial formula for permutations: nPr = n! / (n - r)!. Here n = 10 and r = 4, so 10P4 = 10! / 6!. The value 10! equals 3628800 and 6! equals 720. Dividing 3628800 by 720 gives 5040, which matches our previous calculation. This consistency provides strong confirmation that we have counted correctly.

Why Other Options Are Wrong:
Values such as 4050, 3600 and 1200 do not equal 10P4. They typically arise from incomplete multiplication (for example, stopping at 10 * 9 * 8) or mixing combinations with permutations. For instance, 10C4 * 4! simplifies to 10P4, but if someone uses only 10C4, they will get a much smaller value. Only 5040 correctly represents the number of ordered 4 letter arrangements formed from 10 distinct letters without repetition.

Common Pitfalls:
A frequent error is to confuse this situation with combinations and use 10C4, which counts only unordered selections of letters. Another mistake is to allow repetition unintentionally, which would change the counting completely. Some learners also miscount the number of distinct letters in LOGARITHMS and treat it as 9 or 8 letters. Always verify the distinct letters and decide whether order matters before choosing between permutations and combinations.

Final Answer:
The number of different 4 letter words that can be formed from the letters of LOGARITHMS without repetition is 5040.