How many ways can 5 different letters be arranged if 2 letters must be together?

How many ways can 5 different letters be arranged if 2 letters must be together?

Correct Answer: 48

Step 1: Identify the 5 different letters. Let's call them A, B, C, D, and E.
Step 2: Choose the 2 letters that must be together. For example, let's say we choose A and B.
Step 3: Treat the 2 letters (A and B) as one single unit or block. Now, instead of 5 letters, we have 4 units to arrange: (AB), C, D, and E.
Step 4: 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 5: Calculate 4! = 4 × 3 × 2 × 1 = 24.
Step 6: Now, within the block (AB), the letters A and B can be arranged in 2 ways: AB or BA. This is 2!.
Step 7: Calculate 2! = 2 × 1 = 2.
Step 8: Multiply the number of arrangements of the 4 units by the arrangements of the 2 letters in the block: 24 (from Step 5) × 2 (from Step 7) = 48.
Step 9: Therefore, the total number of ways to arrange the 5 letters with 2 letters together is 48.

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