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

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

Correct Answer: 56

Step 1: Identify the total number of letters in the word 'COMPUTER'. There are 8 letters: C, O, M, P, U, T, E, R.
Step 2: Understand that we want to choose 3 letters from these 8 letters.
Step 3: Use the combination formula C(n, r) which is used to find the number of ways to choose r items from n items without regard to the order of selection. The formula is C(n, r) = n! / (r! * (n - r)!).
Step 4: In our case, n = 8 (total letters) and r = 3 (letters we want to choose).
Step 5: Plug the values into the formula: C(8, 3) = 8! / (3! * (8 - 3)!) = 8! / (3! * 5!).
Step 6: Calculate 8! = 8 × 7 × 6 × 5! (we can cancel 5! in the numerator and denominator).
Step 7: Now we have C(8, 3) = (8 × 7 × 6) / (3 × 2 × 1).
Step 8: Calculate the numerator: 8 × 7 × 6 = 336.
Step 9: Calculate the denominator: 3 × 2 × 1 = 6.
Step 10: Divide the numerator by the denominator: 336 / 6 = 56.
Step 11: Therefore, the number of different ways to choose 3 letters from the word 'COMPUTER' is 56.

Combinations – The concept of combinations is used to determine the number of ways to choose a subset of items from a larger set without regard to the order of selection.
Counting Principles – Understanding how to apply counting principles, such as the combination formula C(n, r) = n! / (r!(n-r)!), is essential for solving problems involving selections.