in how many ways can the letters of the word apple be arranged?

In how many ways can the letters of the word “APPLE” be arranged?
Answer: 60 distinct arrangements.

Understanding the word “APPLE”

The word APPLE has 5 letters:

• A – 1 time
• P – 2 times
• L – 1 time
• E – 1 time

So, we are arranging 5 letters where one letter (P) is repeated twice.

The permutation formula with repeated letters

When letters repeat, we use:
Number of arrangements = n!k1! k2!…\dfrac{n!}{k_1!,k_2!\dots}k1​!k2​!…n!​
where nnn is the total number of letters, and each kik_iki​ is how many times a particular letter repeats.

For APPLE:

• Total letters n=5n=5n=5
• P repeats twice, so k1=2k_1=2k1​=2

So:

Arrangements=5!2!\text{Arrangements}=\dfrac{5!}{2!}Arrangements=2!5!​

Compute:

• 5!=5×4×3×2×1=1205!=5\times 4\times 3\times 2\times 1=1205!=5×4×3×2×1=120
• 2!=2×1=22!=2\times 1=22!=2×1=2

Then:

1202=60\dfrac{120}{2}=602120​=60

So, there are 60 distinct permutations of the letters of the word APPLE.

Quick story-style intuition

Imagine writing all possible 5-letter strings made from A, P, P, L, E as if all letters were different; you’d get 120 arrangements. But since the two P’s are indistinguishable, every arrangement is counted twice in that list—once for each way of swapping the two P positions—so you divide by 2, leaving 60 truly different words.

HTML mini-table: key facts

html

<table>
  <tr>
    <th>Item</th>
    <th>Value</th>
  </tr>
  <tr>
    <td>Word</td>
    <td>APPLE</td>
  </tr>
  <tr>
    <td>Total letters (n)</td>
    <td>5</td>
  </tr>
  <tr>
    <td>Repeated letter</td>
    <td>P (2 times)</td>
  </tr>
  <tr>
    <td>Formula used</td>
    <td>5! / 2!</td>
  </tr>
  <tr>
    <td>Final number of arrangements</td>
    <td>60</td>
  </tr>
</table>

TL;DR: The letters of the word “APPLE” can be arranged in 60 different ways.

Information gathered from public forums or data available on the internet and portrayed here.