Question

How many three-letter words with or without meaning, can be formed out of the letters of the word LOGARITHMS if repetition of letters is not allowed?
A. 720
B. 420
C. 5040
D. none of these

Answer 1

Hint: Here, we can first try to find out the total number of letters from the given word. Then we can try to find the permutation expression to find the ways. Then, we will solve the expression and get our answer.

Complete step by step solution:
First, we will find the total number of letters in the word and we get:
We know that the letters in the word LOGARITHMS are all different words. So, the total number of letters in LOGARITHMS= \[10\]
Now, according to the question, we want to find for \[3\]letter word so our expression will be \[{}^{10}{C_3}\].
This means that we can get our \[3\]letter from \[10\] in \[{}^{10}{C_3}\]ways.
Now, these \[3\]letters can also be arranged in the form of \[3!\]ways.
When we start calculating the total number of words, then we get:
\[ = 3!\times {}^{10}{C_3}\]
The above expression can be written as:
\[ = 3!\times 10!(7! \times 3!)\]
\[ = 10!7!\]
\[ = 10 \times 9\times 8\]
\[ = 720\]

Therefore, the answer is \[720\]. So, we can say that \[720\] words can be made now. So, the correct option for the above question is the option B.

Note:
We have another method also. We can use the permutation method. We can assume and take \[10\] ways for the first letter. The same way, we can take \[9\] ways for the second letter and \[8\] ways for the third letter. Then we can calculate the total number of words by multiplying them and get the answer as \[720\].