Let $S = \{1, 2, 3, ….., 100\}$. The number of non-empty subsets $A$ of $S$ such that the product of elements in $A$ is even is

If $\mathrm{S}=\{\mathrm{a} \in \mathrm{R}:|2 \mathrm{a}-1|=3[\mathrm{a}]+2\{\mathrm{a}\}\}$, where $[\mathrm{t}]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents the fractional part of $t$, then $72 \sum_{\mathrm{a} \in \mathrm{S}} \mathrm{a}$ is equal to....................

If $X = \{ {4^n} - 3n - 1:n \in N\} $ and $Y = \{ 9(n - 1):n \in N\} ,$ then $X \cup Y$ is equal to

Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers satisfying the condition $x^2-y^2=12345678$. Then,

If $\mathrm{A}=\{\mathrm{x} \in {R}:|\mathrm{x}-2|>1\}, \mathrm{B}=\left\{\mathrm{x} \in {R}: \sqrt{\mathrm{x}^{2}-3}>1\right\}$, $\mathrm{C}=\{\mathrm{x} \in {R}:|\mathrm{x}-4| \geq 2\}$ and ${Z}$ is the set of all integers, then the number of subsets of the set $(A \cap B \cap C)^{c} \cap {Z}$ is .... .

Let $A_1, A_2, \ldots \ldots, A_m$ be non-empty subsets of $\{1,2,3, \ldots, 100\}$ satisfying the following conditions:

$1.$ The numbers $\left|A_1\right|,\left|A_2\right|, \ldots,\left|A_m\right|$ are distinct.

$2.$ $A_1, A_2, \ldots, A_m$ are pairwise disjoint.(Here $|A|$ donotes the number of elements in the set $A$ )Then, the maximum possible value of $m$ is