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

Options

A. 10080
B. 2520
C. 1260
D. 5040

Answer

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

### Step 1: Count the total number of letters
The word "LEARNER" consists of 7 letters: L, E, A, R, N, E, R.

### Step 2: Identify the frequency of each letter
- L: 1 time
- E: 2 times
- A: 1 time
- R: 2 times
- N: 1 time

### Step 3: Use the formula for arrangements of letters
The formula to find 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 repeated letters.

### Step 4: Plug in the values
Here, \( n = 7 \) (total letters), and the frequencies of the repeated letters are:
- E appears 2 times,
- R appears 2 times.

So, the formula becomes:

\[ \text{Number of arrangements} = \frac{7!}{2! \times 2!} \]

### Step 5: Calculate the factorials
Now, we calculate \( 7! \) and \( 2! \):

- \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \)
- \( 2! = 2 \times 1 = 2 \)

### Step 6: Substitute the values into the formula
Now substituting these values into the formula:

\[ \text{Number of arrangements} = \frac{5040}{2 \times 2} = \frac{5040}{4} = 1260 \]

### Final Answer
Thus, the number of different ways to arrange the letters of the word "LEARNER" is **1260**.

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