10 books are placed at random in a shelf. The probability that a pair of books will always be together is?
Answer: 1/5
Explanation
To find the probability that a pair of books will always be together, we need to count the number of arrangements where the pair is together and divide it by the total number of arrangements.
Let's consider the pair of books as a single unit. Then, we have 9 units (the pair and the remaining 8 books) to arrange on the shelf. The number of ways to arrange these 9 units is 9!.
However, within the pair, the two books can be arranged in 2! ways (either book can be on the left or right).
So, the total number of arrangements where the pair is together is 9! × 2!.
Now, we divide the number of favorable arrangements (where the pair is together) by the total number of arrangements:
Probability = (9! × 2!) / 10!
= (9! / 10!) × 2
= (1/10) × 2
= 1/5
