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

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

Options:

  1. 48
  2. 60
  3. 120
  4. 24

Correct Answer: 60

Solution: 

Treat the two specific letters as one unit. Then, we have 4 units to arrange: 4! = 24. The two letters can be arranged in 2! = 2 ways. Total = 24 * 2 = 48.

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

Practice Questions

Q1
How many ways can 5 different letters be arranged such that two specific letters are always together?
  1. 48
  2. 60
  3. 120
  4. 24

Questions & Step-by-Step Solutions

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.
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