How many permutations to the word APPLE are there?

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To determine the number of distinct permutations of the word **APPLE**, follow these steps:

1. **Count the Total Letters:**
   
   The word "APPLE" has **5** letters.

2. **Identify and Count Repeating Letters:**
   
   - **A**: 1
   - **P**: 2
   - **L**: 1
   - **E**: 1
   
   The letter **P** repeats **2** times.

3. **Use the Permutation Formula for Repeating Items:**
   
   The formula to calculate the number of distinct permutations when there are repeating letters is:

$$
   \text{Number of permutations} = \frac{n!}{n_1! \times n_2! \times \cdots \times n_k!}
   $$
   
   Where:
   - $$ n $$ is the total number of letters.
   - $$ n_1, n_2, \ldots, n_k $$ are the frequencies of the repeating letters.

4. **Apply the Values:**
   
   For "APPLE":
   - Total letters, $$ n = 5 $$
   - The letter **P** repeats $$ n_1 = 2 $$ times.

Plugging into the formula:

$$
   \text{Number of permutations} = \frac{5!}{2!} = \frac{120}{2} = 60
   $$

**Answer:** There are **60** distinct permutations of the word "APPLE".
There are 60 distinct permutations of the word "APPLE".

How many permutations to the word
APPLE are there?

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