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

Options

A. 6020
B. 5040
C. 6220
D. 6720

Answer

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

### Step 1: Count the total letters in the word
The word "DIVISION" consists of 8 letters.

### Step 2: Identify the repetitions of letters
In the word "DIVISION":
- The letter 'I' appears 3 times.
- The letters 'D', 'V', 'S', 'O', and 'N' each appear 1 time.

### Step 3: Use the formula for permutations of multiset
The formula for the number of arrangements of letters in a word with 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 letters,
- \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

For "DIVISION":
- Total letters \( n = 8 \)
- Repetitions: \( p_1 = 3! \) (for 'I')

### Step 4: Substitute the values into the formula
\[ \text{Number of arrangements} = \frac{8!}{3!} \]

### Step 5: Calculate the factorials
First, calculate \( 8! \) and \( 3! \):
- \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)

### Step 6: Perform the division
Now, divide \( 8! \) by \( 3! \):
\[ \frac{40320}{6} = 6720 \]

### Conclusion
The total number of different ways the letters of the word "DIVISION" can be arranged is **6720**.

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