How many different ways can 3 letters be chosen from the word 'APPLE'?

How many different ways can 3 letters be chosen from the word 'APPLE'?

Correct Answer: 10

Step 1: Identify the letters in the word 'APPLE'. The letters are A, P, P, L, E.
Step 2: Count the total number of letters. There are 5 letters in total.
Step 3: Note that the letter 'P' is repeated. This means we have to consider combinations carefully.
Step 4: Use the combination formula C(n, r) where n is the total number of letters and r is the number of letters to choose. Here, n = 5 and r = 3.
Step 5: Calculate C(5, 3). This is done using the formula C(n, r) = n! / (r! * (n - r)!).
Step 6: Calculate C(5, 3) = 5! / (3! * (5 - 3)!) = 5! / (3! * 2!) = (5 * 4) / (2 * 1) = 10.
Step 7: Since 'P' is repeated, we need to check if we counted any duplicates. The combinations are: APP, APL, APE, PPL, PPE, PLE, APL, APE, PLE, and PLE.
Step 8: After checking for duplicates, we confirm that the total number of unique combinations of 3 letters from 'APPLE' is indeed 10.

Combinations with Repetition – The problem involves choosing letters from a set where some elements are repeated, requiring careful consideration of unique combinations.
Counting Distinct Arrangements – Understanding how to count distinct arrangements when letters are repeated, as in the case of 'P' in 'APPLE'.