Find the number of 4 letter words, with or without meaning, which can be formed out of the letters of the word ROSE, where the repetition of the letters is not allowed.

Step 1: Identify the given letters and constraints

The problem asks us to form 4-letter words using the letters from the word 'ROSE'. The letters available are R, O, S, E. We need to form words of length 4, and repetition of letters is not allowed. This means each letter can be used only once.

Step 2: Determine the method for counting arrangements

Since the order in which the letters are arranged matters (e.g., 'ROSE' is different from 'SORE'), and repetition is not allowed, this is a problem of permutations. The formula for permutations of $n$ distinct items taken $r$ at a time is $P(n, r) = \frac{n!}{(n-r)!}$.

Step 3: Substitute values into the permutation formula

In this problem, we have $n=4$ (four distinct letters: R, O, S, E) and we are forming words of length $r=4$. Substituting these values into the permutation formula, we get $P(4, 4) = \frac{4!}{(4-4)!}$.

Step 4: Calculate the factorial values

First, simplify the denominator: $(4-4)! = 0!$. By definition, $0! = 1$. Next, calculate $4!$, which is $4 \times 3 \times 2 \times 1 = 24$.

Step 5: Compute the final number of words

Now, substitute the calculated factorial values back into the permutation formula: $P(4, 4) = \frac{24}{1} = 24$. Therefore, 24 different 4-letter words can be formed.

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