Question

$n$ different books $(n\ge 3)$ are put at random in a shelf. Among these books there is a particular book '$A$' and a particular book $B$. The probability that there are exactly '$r$' books between $A$ and $B$ is-

• A. $\frac{2}{n(n-1)}$
• B. $\frac{2(n-r-1)}{n(n-1)}$
• C. $\frac{2(n-r-2)}{n(n-1)}$
• D. $\frac{(n-1)}{n(n-1)}$

Answer 1

The correct option is B $\frac{2(n-r-1)}{n(n-1)}$

If there are r books between 'A' and 'B' then the block of books with books A and B at the ends has length $r+2$. Consequently the block must begin in one of the first
$n-(r+2)+1=n-r-1$ positions
There are $2$ ways to choose whether book A or book B is at the left end of the block.
There are $(n-2)!$ ways to arrange the remaining books in order.
Hence the probability that there are exactly r books between A and B is
$\frac{2(n-r-1)(n-2)!}{n!}$
$=\frac{2(n-r-1)(n-2)!}{n(n-1)(n-2)!}$
$=\frac{2(n-r-1)}{n(n-1)}$
Ans:$=\frac{2(n-r-1)}{n(n-1)}$