In how many ways can the letters of the word 'CHEESE' be arranged?

Explanation

To find the number of ways to arrange the letters of the word "CHEESE", we can follow these steps:

### Step 1: Count the total number of letters
The word "CHEESE" consists of 6 letters: C, H, E, E, S, E.

### Step 2: Identify the repeated letters
In the word "CHEESE", the letter E appears 3 times.

### Step 3: Use the formula for permutations of multiset
The formula for the number of arrangements of letters where some letters are repeated is given by:

\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

Where:
- \( n \) is the total number of letters,
- \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

In this case:
- \( n = 6 \) (total letters),
- The letter E is repeated 3 times, so \( p_1 = 3! \) (for the letter E).

### Step 4: Calculate the factorial values
Now, we can calculate the factorials:
- \( 6! = 720 \)
- \( 3! = 6 \)

### Step 5: Substitute the values into the formula
Now we can substitute these values into the formula:

\[ \text{Number of arrangements} = \frac{6!}{3!} = \frac{720}{6} = 120 \]

### Conclusion
Thus, the number of ways to arrange the letters of the word "CHEESE" is **120**.

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