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?
48
60
72
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.
