How many 4-letter words, with or without meaning, can be formed out of the letters in the word LOGARITHMS, if repetition of letters is not allowed ?

Explanation

To solve the problem of how many 4-letter words can be formed from the letters in the word "LOGARITHMS" without repetition of letters, we can follow these steps:

### Step 1: Identify the unique letters in the word "LOGARITHMS"
The word "LOGARITHMS" consists of the following unique letters: L, O, G, A, R, I, T, H, M, S.

### Step 2: Count the number of unique letters
The total number of unique letters in "LOGARITHMS" is 10.

### Step 3: Determine the number of ways to choose 4 letters from the 10 unique letters
Since repetition of letters is not allowed, we need to choose 4 letters from the 10 available letters. The number of ways to choose 4 letters from 10 can be calculated using the combination formula:

\[ \text{Number of ways to choose 4 letters} = \binom{10}{4} \]

### Step 4: Calculate the combination
Using the combination formula:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

For our case, \( n = 10 \) and \( r = 4 \):

\[ \binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4! \cdot 6!} \]

Calculating this gives:

\[ = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = \frac{5040}{24} = 210 \]

### Step 5: Determine the number of arrangements of the chosen letters
After choosing 4 letters, we can arrange these 4 letters in different ways. The number of arrangements of 4 letters is given by \( 4! \):

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 6: Calculate the total number of 4-letter words
To find the total number of 4-letter words, we multiply the number of ways to choose the letters by the number of arrangements:

\[ \text{Total 4-letter words} = \binom{10}{4} \times 4! = 210 \times 24 \]

Calculating this gives:

\[ 210 \times 24 = 5040 \]

### Final Answer
Thus, the total number of 4-letter words that can be formed from the letters in "LOGARITHMS" without repetition is **5040**.

---