How many ways can 5 different letters be selected from the alphabet?

How many ways can 5 different letters be selected from the alphabet?

Step 1: Understand that we have 26 letters in the alphabet.
Step 2: We want to select 5 different letters from these 26 letters.
Step 3: Recognize that the order in which we select the letters does not matter, so we will use combinations.
Step 4: The formula for combinations is C(n, r) = n! / (r! * (n - r)!), where n is the total number of items, r is the number of items to choose, and '!' denotes factorial.
Step 5: In our case, n = 26 (total letters) and r = 5 (letters to choose).
Step 6: Plug the values into the formula: C(26, 5) = 26! / (5! * (26 - 5)!)
Step 7: Simplify the expression: C(26, 5) = 26! / (5! * 21!)
Step 8: Calculate the factorials: 26! = 26 × 25 × 24 × 23 × 22 × 21!, and 5! = 5 × 4 × 3 × 2 × 1 = 120.
Step 9: Substitute back into the equation: C(26, 5) = (26 × 25 × 24 × 23 × 22) / 120.
Step 10: Perform the multiplication: 26 × 25 × 24 × 23 × 22 = 7893600.
Step 11: Divide by 120: 7893600 / 120 = 65780.
Step 12: Conclude that there are 65780 ways to select 5 different letters from the alphabet.

Combinatorics – The question tests the understanding of combinations, specifically how to calculate the number of ways to choose a subset of items from a larger set without regard to the order of selection.