In how many ways can 5 different letters be arranged if 2 letters are always together? (2019)

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Q: In how many ways can 5 different letters be arranged if 2 letters are always together? (2019)

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

Steps: 8

Step 1: Identify the 5 different letters. Let's call them A, B, C, D, and E.
Step 2: Since 2 letters must always be together, let's group those 2 letters. For example, let's say we group A and B together. We can treat this group as one unit.
Step 3: 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, remember that the 2 letters (A and B) can be arranged among themselves. There are 2! ways to arrange A and B.
Step 7: Calculate 2! = 2 × 1 = 2.
Step 8: Finally, multiply the number of arrangements of the 4 units by the arrangements of the 2 letters: 4! × 2! = 24 × 2 = 48.