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

1 question

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Q: How many ways can 5 different letters be arranged if 2 letters are always together?

Solution: Treat the 2 letters as one unit. So, we have 4 units to arrange: 4! * 2! = 48.

Steps: 9

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 always 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 letters within 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 A and B always together is 48.