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

Options

A. 910
B. 2520
C. 5040
D. 840

Answer

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

### Step 1: Identify the total number of letters
The word "PADDLED" consists of 7 letters: P, A, D, D, L, E, D.

### Step 2: Count the frequency of each letter
- P: 1
- A: 1
- D: 3
- L: 1
- E: 1

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

\[ \text{Number of arrangements} = \frac{n!}{n_1! \cdot n_2! \cdot n_3! \cdots} \]

Where:
- \( n \) is the total number of letters,
- \( n_1, n_2, n_3, \ldots \) are the frequencies of the repeated letters.

In our case:
- \( n = 7 \) (total letters)
- The letter D is repeated 3 times, while P, A, L, and E are each repeated once.

### Step 4: Substitute the values into the formula
\[ \text{Number of arrangements} = \frac{7!}{3! \cdot 1! \cdot 1! \cdot 1! \cdot 1!} \]

### Step 5: Calculate the factorials
- \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)
- \( 1! = 1 \) (for P, A, L, and E)

### Step 6: Perform the division
\[ \text{Number of arrangements} = \frac{5040}{6} = 840 \]

### Final Answer
The total number of different ways the letters of the word "PADDLED" can be arranged is **840**.

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