If the integers m and n are chosen at random from integers 1 to 100 with replacement, then the probability that a number of the form 7 $^{m}$ ? is divisible by 5 equals

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Two numbers X and Y are chosen at random (without replacement) from the set {1,2,...5N]. The probability that X4−Y4 is divisible by 5 is

The correct option is C 17N−55(5N−1)x4−y4 is divisible by 5, if and only if both x and y are divisible by 5 or none of them is divisible by 5. Thus, favourable ways is NC2+4NC2.Hence required probability =17N−55(5N−1)

Two numbers $X$ and $Y$ are chosen at random (without replacement) from the set ${1, 2, . . . 5 N]$ . The probability that $X^{4} - Y^{4}$ is divisible by $5$ is

The correct option is C $\frac{17 N - 5}{5 (5 N - 1)}$
$x^{4} - y^{4}$ is divisible by 5, if and only if both $x$ and $y$ are divisible by 5 or none of them is divisible by 5.
Thus, favourable ways is $^{N} C_{2} +^{4 N} C_{2}$ .
Hence required probability $= \frac{17 N - 5}{5 (5 N - 1)}$