How many ways can 5 different letters be arranged such that two specific letters

How many ways can 5 different letters be arranged such that two specific letters are always together?

How many ways can 5 different letters be arranged such that two specific letters are always together?

Step 1: Identify the two specific letters that need to be together. Let's call them A and B.
Step 2: Treat the two letters A and B as one single unit or block. Now, instead of 5 letters, we have 4 units to arrange: the block (AB) and the other 3 letters.
Step 3: Calculate the number of ways to arrange these 4 units. The formula for arranging n units is n!. So, we calculate 4! (which is 4 factorial).
Step 4: Calculate 4! = 4 × 3 × 2 × 1 = 24. This means there are 24 ways to arrange the 4 units.
Step 5: Now, consider the arrangement of the two letters A and B within their block. They can be arranged in 2 ways: AB or BA.
Step 6: Multiply the number of arrangements of the 4 units (24) by the number of arrangements of A and B (2). So, 24 × 2 = 48.
Step 7: The final answer is that there are 48 different ways to arrange the 5 letters with A and B always together.

Permutations with Restrictions – This concept involves arranging items with specific conditions, such as keeping certain items together.
Factorial Calculations – Understanding how to calculate permutations using factorials is crucial for solving arrangement problems.