Out of first 5 consecutive natural numbers, if two different numbers x and y are chosen at random, then the…
Out of first 5 consecutive natural numbers, if two different numbers x and y are chosen at random, then the probability that $x^4-y^4$ is divisible by 5 is
$\frac{2}{5}$
$\frac{4}{5}$
$\frac{3}{5}$
$\frac{1}{5}$
Solution
$\because$ We know, if two numbers $x$ and $y$ are choosen at random without replacement from the set $\{1,2,3,4, \ldots .$. $5 K\}$, then the probability that $x^4-y^4$ is divisible by 5 is $\frac{17 K-5}{5(5 K-1)}$
$\because \text { Given set }=\{1,2,3,4,5\} \Rightarrow K=1$
So, required probability $=\frac{17 \times 1-5}{5(5 \times 1-1)}=\frac{12}{5 \times 4}=\frac{3}{5}$.