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

Options

A. 840
B. 5040
C. 2520
D. 1680

Explanation

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

### Step 1: Count the total number of letters
The word "MARKERS" consists of 7 letters: M, A, R, K, E, R, S.

### Step 2: Identify any repeating letters
In the word "MARKERS", the letter 'R' appears twice.

### Step 3: Use the formula for permutations of multiset
The formula for the number of arrangements of n items where there are repetitions 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 items, and \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeating items.

### Step 4: Apply the formula
Here, \( n = 7 \) (total letters) and the letter 'R' repeats 2 times. Thus, we have:

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

### Step 5: Calculate the factorials
First, we calculate \( 7! \):

\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]

Now, calculate \( 2! \):

\[ 2! = 2 \times 1 = 2 \]

### Step 6: Substitute the values into the formula
Now substitute the values into the formula:

\[ \text{Number of arrangements} = \frac{5040}{2} = 2520 \]

### Final Answer
Thus, the letters of the word "MARKERS" can be arranged in **2520 different ways**.

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