Question

Two numbers $X$ and $Y$ are chosen at random (without replacement) from the set $1,2,...,5N$. The probability that $X^n\sim Y^n$ is divisible by $5$ is

• A. $\frac{N-1}{5N-1}$
• B. $\frac{4(4N-1)}{5(5N-1)}$
• C. $\frac{17N-5}{5(5N-1)}$
• D. None of these

Answer 1

The correct option is A $\frac{N-1}{5N-1}$

The number of ways of selecting $X$ and $Y$ is ${}^{5N}{C}_2.$
Arranging the numbers $1,2,3,...5N$ is five rows
$1,6,11,...,5N-4$
$2,7,12,...,5N-3$
$3,8,13,...,5N-2$
$4,9,14,...,5N-1$
$5,10,15,...,5N$
we see that $X^n-Y^n$ will be divisible by $5$ if both $X$ and $Y$ lie in same row.
Thus the number of favourable ways is $5^N{C}_2.$
Hence required probability $=\frac{5^N{C}_2}{{}^{5N}{C}_2}=\frac{5N(N-1)}{5N(5N-1)}=\frac{N-1}{(5N-1)}$