How many ways can 4 letters be chosen from the word 'COMBINATION'?

How many ways can 4 letters be chosen from the word 'COMBINATION'?

Step 1: Identify the total number of letters in the word 'COMBINATION'. There are 11 letters.
Step 2: Understand that we want to choose 4 letters from these 11 letters.
Step 3: Use the combination formula, which is written as nCr, where n is the total number of items to choose from and r is the number of items to choose. Here, n = 11 and r = 4.
Step 4: The combination formula is nCr = n! / (r! * (n - r)!), where '!' denotes factorial, which is the product of all positive integers up to that number.
Step 5: Calculate 11C4 using the formula: 11C4 = 11! / (4! * (11 - 4)!) = 11! / (4! * 7!).
Step 6: Simplify the calculation: 11! = 11 × 10 × 9 × 8 × 7!, so we can cancel 7! in the numerator and denominator.
Step 7: Now we have 11C4 = (11 × 10 × 9 × 8) / (4 × 3 × 2 × 1).
Step 8: Calculate the numerator: 11 × 10 × 9 × 8 = 7920.
Step 9: Calculate the denominator: 4 × 3 × 2 × 1 = 24.
Step 10: Divide the numerator by the denominator: 7920 / 24 = 330.
Step 11: Therefore, the number of ways to choose 4 letters from the word 'COMBINATION' is 330.

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.
Handling Repeated Letters – The word 'COMBINATION' contains repeated letters (two 'O's and two 'I's), which complicates the counting of unique combinations.