Step 1: Count the total distinct letters

The word LOGARITHMS has 10 distinct letters:
L, O, G, A, R, I, T, H, M, S (no repetitions).

Step 2: Select 4 letters

Since we need to form a 4-letter word, we first choose 4 letters out of the 10.
The number of ways to choose 4 letters from 10 is:

Ways to choose 4 letters=(104)=10!4!(10−4)!\text{Ways to choose 4 letters} = \binom{10}{4} = \frac{10!}{4!(10-4)!}Ways to choose 4 letters=(410​)=4!(10−4)!10!​ =10!4!6!=10×9×8×74×3×2×1=210= \frac{10!}{4!6!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210=4!6!10!​=4×3×2×110×9×8×7​=210
Step 3: Arrange the 4 chosen letters

Since the arrangement matters (as we are forming words), we arrange the selected 4 letters in:

4!=4×3×2×1=244! = 4 \times 3 \times 2 \times 1 = 244!=4×3×2×1=24
Step 4: Calculate the total number of words
Total words=210×24=5040\text{Total words} = 210 \times 24 = 5040Total words=210×24=5040
Final Answer:

The number of 4-letter words that can be formed from LOGARITHMS, without repeating letters, is 5,040. 🎉