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

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

Correct Answer: 165

Step 1: Identify the total number of letters in the word 'COMBINATION'. There are 11 letters.
Step 2: Determine that we want to choose 3 different letters from these 11 letters.
Step 3: Use the combination formula C(n, r) = n! / (r! * (n - r)!), where n is the total number of items (11 letters) and r is the number of items to choose (3 letters).
Step 4: Plug in the values: C(11, 3) = 11! / (3! * (11 - 3)!) = 11! / (3! * 8!).
Step 5: Simplify the factorials: C(11, 3) = (11 * 10 * 9) / (3 * 2 * 1).
Step 6: Calculate the numerator: 11 * 10 * 9 = 990.
Step 7: Calculate the denominator: 3 * 2 * 1 = 6.
Step 8: Divide the numerator by the denominator: 990 / 6 = 165.
Step 9: Conclude that there are 165 different ways to choose 3 letters from the word 'COMBINATION'.

Combinatorics – The question tests the understanding of combinations, specifically how to choose a subset of items from a larger set without regard to the order of selection.
Counting Distinct Elements – The problem requires recognizing that the letters in 'COMBINATION' include duplicates, which affects the total number of distinct letters available for selection.