Question

The numbers $1,2,3,...,n$ are arranged in a random order. The probability that the digits $1,2,3,...k(n>k)$ appear as neighbors is

• A. $\frac{(n-k)!}{n!}$
• B. $\frac{n-k+1}{{}^n{C}_k}$
• C. $\frac{n-k}{{}^n{C}_k}$
• D. $\frac{k!}{n!}$

Answer 1

The correct option is B $\frac{n-k+1}{{}^n{C}_k}$

The number $1,2,3,...,n$ can be arranged in a row in $n!$ ways.
The total number of ways in which the digits $1,2,...,k(k<n)$ occur together is $k!(n-k+1)!$
Hence, required probability $=\frac{k!(n-k+1)!}{n!}=\frac{n-k+1}{{}^n{C}_k}$