How many $3$ -letter words with or without meaning, can be formed out of the letters of the word 'LOGARITHMS', if repetition of letters is not allowed?

The given word is 'LOGARITHMS'.
Let us count the number of unique letters in this word:

$$\text{L, O, G, A, R, I, T, H, M, S}$$

There are $10$ distinct letters in total.
We need to form a $3$ -letter word without repeating any letters. This is a problem of finding the number of permutations of $10$ distinct objects taken $3$ at a time:

$${}^{10}{\text{P}}_3=\frac{10!}{(10-3)!}=\frac{10!}{7!}=10\times 9\times 8=720$$

Alternatively, using the fundamental counting principle:

* The first position can be filled in $10$ ways.
* The second position can be filled in $9$ ways (as repetition is not allowed).
* The third position can be filled in $8$ ways.

$$\text{Total number of words}=10\times 9\times 8=720$$