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

Options

A. 64
B. 13440
C. 6720
D. 20160

Answer

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

1. **Count the total number of letters**: The word "SEQUENCE" has 8 letters in total: S, E, Q, U, E, N, C, E.

2. **Identify the repeating letters**: In the word "SEQUENCE", the letter 'E' appears 3 times.

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.

4. **Substituting the values**: Here, \( n = 8 \) (total letters), and the frequency of 'E' (which is repeated) is 3. Therefore, we have:
\[ \text{Number of arrangements} = \frac{8!}{3!} \]

5. **Calculate \( 8! \) and \( 3! \)**:
- \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)

6. **Divide \( 8! \) by \( 3! \)**:
\[ \text{Number of arrangements} = \frac{40320}{6} = 6720 \]

7. **Conclusion**: The total number of different ways to arrange the letters of the word "SEQUENCE" is **6720**.