How many four letter words with or without meaning that can be formed by using the letters of the word “LOGARITHMS” such that the letter R is always included? (Repetition is not allowed).

Hint: Here, we will be using permutation to find out the words with or without meaning that can be formed by using the letters of the word “LOGARITHMS”. Firstly we have to calculate the total number of distinct letters in the word “LOGARITHMS”. As letter R is always included in the four letter word therefore we have to use the permutation for the selection of the remaining three letters of the word. After that we will arrange those four letters within the word to find out the number of four letter words with or without meaning that can be formed.

Complete step-by-step answer:
The given word which is used to form a four-letter word is “LOGARITHMS”.
The total number of distinct letters in the given word is 10.
To form a four-letter word with or without meaning such that the letter R is always included and repetition is not allowed, we will use permutation to find out the rest of the three letters of the word.
We have 9 letters remaining from the given word from which we have to select only 3 letters.
So out of 9 letters, we can select any letter as one of the three remaining letters of the four-letter word. As repetition is not allowed, therefore, now out of the remaining 8 letters we can select any letter as one of the two remaining letters of the four-letter word. Now out of the remaining 7 letters, we can select any letter as the last letters of the four-letter word.
Number of ways of selecting the remaining three letters of the four-letter word from the 9 letters
\[ \Rightarrow {}^9{{\rm{P}}_3} = 9 \times 8 \times 7 = 504\]Ways
Now, we have to arrange those four letters within the word.
\[ \Rightarrow 4! = 4 \times 3 \times 2 \times 1 = 24\]
Therefore, the total number of four-letter words with or without meaning that can be formed \[ = 504 \times 24 = 12096\]
Hence, 12096 number of four-letter words with or without meaning that can be formed by using the letters of the word “LOGARITHMS” such that the letter R is always included.

Note: Permutations may be defined as the different ways in which a collection of items can be arranged. For example: The different ways in which the numbers 1, 2 and 3 can be grouped together, taken all at a time, are \[123,{\rm{ }}132,{\rm{ }}213{\rm{ }}231,{\rm{ }}312,{\rm{ }}321\].
So, Number of permutations of n things, taken r at a time, denoted by \[{}^{\rm{n}}{{\rm{P}}_{\rm{r}}} = \dfrac{{{\rm{n}}!}}{{{\rm{(n - r)}}!}}\]
Combinations may be defined as the various ways in which objects from a set may be selected. For example: The different selections possible from the numbers 1, 2, 3 taking 2 at a time, are \[{\rm{12, 23\, and\, 31}}\]
So, Number of combinations possible from n group of items, taken r at a time, denoted by \[^{\rm{n}}{{\rm{C}}_{\rm{r}}} = \dfrac{{{\rm{n}}!}}{{{\rm{r}}!{\rm{(n - r)}}!}}\]