In how many ways can the letters LEADER be arranged?

Analysis

The question asks for the number of ways to arrange the letters in the word LEADER. This is a problem of permutations of objects where some are identical.

Concept

When arranging $n$ objects where some objects are repeated, we use the formula:

Where:

• $n$ is the total number of letters.
• $n_1, n_2, \dots$ are the frequencies of the repeated letters.

Step-by-Step Solution

1. Count the total number of letters ($n$): The word LEADER has 6 letters. So, $n = 6$.

2. Identify the frequency of each letter: • L: 1 • E: 2 (appears twice) • A: 1 • D: 1 • R: 1

3. Apply the formula: Since the letter 'E' repeats 2 times, we divide the total permutations ($6!$) by the factorial of the number of repetitions ($2!$): $$\text{Arrangements} = \frac{6!}{2!}$$

4. Calculate the values: • $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$ • $2! = 2 \times 1 = 2$

Conclusion

The total number of ways to arrange the letters is 360.

Exam Tip

Always count the total letters first, then immediately look for repeats. A common mistake is to forget to divide by the factorial of the repeated letters. If you see a word with repeated letters, remember: "Total factorial divided by the factorial of the repeats."