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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 never together?

Options:

  1. 48
  2. 60
  3. 72
  4. 96

Correct Answer: 72

Solution: 

Total arrangements = 5! = 120. Arrangements with the two letters together = 4! * 2! = 48. So, arrangements where they are not together = 120 - 48 = 72.

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 never together?
  1. 48
  2. 60
  3. 72
  4. 96

Questions & Step-by-Step Solutions

How many ways can 5 different letters be arranged such that two specific letters are never together?
  • Step 1: Calculate the total number of arrangements of 5 different letters. This is done using the factorial of 5, which is 5! = 5 × 4 × 3 × 2 × 1 = 120.
  • Step 2: Identify the two specific letters that should not be together. Let's call them A and B.
  • Step 3: Treat the two specific letters (A and B) as a single unit or block. This means we now have 4 units to arrange: (AB), C, D, and E.
  • Step 4: Calculate the arrangements of these 4 units. This is done using the factorial of 4, which is 4! = 4 × 3 × 2 × 1 = 24.
  • Step 5: Since A and B can be arranged within their block in 2 different ways (AB or BA), multiply the arrangements of the 4 units by 2. So, 24 × 2 = 48.
  • Step 6: Now, subtract the arrangements where A and B are together from the total arrangements. This gives us 120 - 48 = 72.
  • Step 7: The final answer is that there are 72 ways to arrange the 5 letters such that A and B are never together.
  • Permutations – The arrangement of objects in a specific order, calculated using factorials.
  • Complementary Counting – A technique where you count the total arrangements and subtract the arrangements that do not meet the criteria.
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