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

Options

A. 2520
B. 2025
C. 5040
D. 4050

Answer

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

### Step 1: Identify the total number of letters
The word "COOKING" consists of 7 letters: C, O, O, K, I, N, G.

### Step 2: Count the frequency of each letter
In "COOKING":
- C appears 1 time
- O appears 2 times
- K appears 1 time
- I appears 1 time
- N appears 1 time
- G appears 1 time

### Step 3: Use the formula for permutations of multiset
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: Substitute the values into the formula
In our case:
- \( n = 7 \) (total letters)
- The letter O is repeated 2 times, so \( p_1 = 2! \)

Thus, the number of arrangements is:
\[ \text{Number of arrangements} = \frac{7!}{2!} \]

### Step 5: Calculate \( 7! \) and \( 2! \)
Calculating \( 7! \):
\[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \]

Calculating \( 2! \):
\[ 2! = 2 \times 1 = 2 \]

### Step 6: Divide \( 7! \) by \( 2! \)
Now we can find the number of arrangements:
\[ \text{Number of arrangements} = \frac{5040}{2} = 2520 \]

### Final Answer
The letters of the word "COOKING" can be arranged in **2520 different ways**.

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