Question

How many four-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) 40
B) 400
C) 5040
D) 2520

Answer 1

Hint: We can take the letters in the given word and count them. Then we can find the permutation of forming 4 letters words with the letters of the given words by calculating the permutation of selecting 4 objects from n objects without replacement, where n is the number of letters in the given word which is obtained by the formula, ${}^n{P}_r={\displaystyle \frac{n!}{\left(n-r\right)!}}$

Complete step by step solution:
We have the word ‘LOGARITHMS’.
We can count the letters. After counting, we can say that there are 10 letters in the given word.
$\Rightarrow n=10$
Now we need to form four letter words from these 10 numbers. As the words can be with or without meaning, we can take all the possible ways of arrangements.
As the repetition is not allowed, we can use the equation ${}^n{P}_r={\displaystyle \frac{n!}{\left(n-r\right)!}}$ where n is the number of objects and r is the number of objects needed to be selected.
So, the number of four-letter words can be formed is given by,
$\Rightarrow {}^{10}{P}_4={\displaystyle \frac{10!}{\left(10-4\right)!}}$
So we have,
$\Rightarrow {}^{10}{P}_4={\displaystyle \frac{10!}{6!}}$
Using properties of factorial, we can write the numerator as,
$\Rightarrow {}^{10}{P}_4={\displaystyle \frac{10\times 9\times 8\times 7\times 6!}{6!}}$
On cancelling common terms we get,
$\Rightarrow {}^{10}{P}_4=10\times 9\times 8\times 7$
Hence we have,
$\Rightarrow {}^{10}{P}_4=5040$
Therefore, the number of four-letter words that can be formed is 5040.

So the correct answer is option C.

Note: Alternate method to solve this problem is by,
We have 10 letters that have to be arranged in four places. It is given that repetition of letters is not allowed. So the letter once used cannot be used again.
So, in the $1^{\text{st}}$ place, we place any of the 10 letters. In the second place we can put any of the remaining 9 letters. In $3^{\text{rd}}$ place we can have any of the 8 letters and in the last place any of the remaining 7 letters can be used.
So, the total arrangement is given by, $10\times 9\times 8\times 7=5040$ .
Therefore, the number of words that can be formed is 5040.