How many permutations of the letters of the word 'APPLE' are there?

Explanation

To find the number of permutations of the letters in the word "APPLE," we can follow these steps:

### Step 1: Identify the total number of letters
The word "APPLE" consists of 5 letters: A, P, P, L, E.

### Step 2: Determine the number of letters and repetitions
In the word "APPLE":
- Total letters (n) = 5
- The letter 'P' is repeated 2 times.

### Step 3: Use the formula for permutations of multiset
The formula for the number of permutations of a word with repeated letters is given by:

\[ \text{Number of permutations} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

where:
- \( n \) is the total number of letters,
- \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

For "APPLE":
- \( n = 5 \) (total letters)
- \( p_1 = 2 \) (for the letter P)

### Step 4: Substitute the values into the formula
Now we can substitute the values into the formula:

\[ \text{Number of permutations} = \frac{5!}{2!} \]

### Step 5: Calculate the factorials
Now we calculate \( 5! \) and \( 2! \):

- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
- \( 2! = 2 \times 1 = 2 \)

### Step 6: Calculate the number of permutations
Now we can calculate the number of permutations:

\[ \text{Number of permutations} = \frac{120}{2} = 60 \]

### Conclusion
Thus, the total number of permutations of the letters in the word "APPLE" is **60**.

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