How many different words can be formed by using all the letters of word "SCHOOL"?

Answer

To find the number of different words that can be formed using all the letters of the word "SCHOOL", we can follow these steps:

### Step-by-Step Solution:

1. **Count the Total Number of Letters**:
The word "SCHOOL" consists of 6 letters.

**Calculation**:
Letters: S, C, H, O, O, L → Total = 6 letters.

2. **Identify Repeating Letters**:
In the word "SCHOOL", the letter 'O' appears twice.

**Repetition Count**:
- O: 2 times

3. **Use the Formula for Permutations with Repeated Elements**:
The formula to find the number of distinct permutations of letters when there are repeating letters is given by:
\[ \text{Number of permutations} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]
where \( n \) is the total number of letters, and \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeating letters.

**In this case**:
- \( n = 6 \) (total letters)
- \( p_1 = 2 \) (for the letter O)

Therefore, the formula becomes:
\[ \text{Number of permutations} = \frac{6!}{2!} \]

4. **Calculate Factorials**:
- Calculate \( 6! \):
\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]
- Calculate \( 2! \):
\[ 2! = 2 \times 1 = 2 \]

5. **Substitute and Simplify**:
Now substitute the values into the formula:
\[ \text{Number of permutations} = \frac{720}{2} = 360 \]

### Final Answer:
The total number of different words that can be formed using all the letters of the word "SCHOOL" is **360**.

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