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

Options

A. 720
B. 180
C. 120
D. 5040

Answer

To find the number of different ways the letters of the word "SERIES" can be arranged, we need to follow these steps:

### Step 1: Count the total number of letters
The word "SERIES" consists of 6 letters: S, E, R, I, E, S.

### Step 2: Identify any repeating letters
In the word "SERIES", the letter 'S' appears 2 times and the letter 'E' also appears 2 times.

### Step 3: Use the formula for permutations of multiset
The formula to calculate the number of arrangements of letters in a word 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 repeating letters.

### Step 4: Apply the formula
Here, \( n = 6 \) (total letters), \( p_1 = 2 \) (for S), and \( p_2 = 2 \) (for E). Thus, we can substitute these values into the formula:

\[ \text{Number of arrangements} = \frac{6!}{2! \times 2!} \]

### Step 5: Calculate the factorials
Now, we calculate the factorials:
- \( 6! = 720 \)
- \( 2! = 2 \)

### Step 6: Substitute and simplify
Now substitute these values back into the equation:

\[ \text{Number of arrangements} = \frac{720}{2 \times 2} = \frac{720}{4} = 180 \]

### Final Answer
Thus, the number of different ways the letters of the word "SERIES" can be arranged is **180**.

---