Let $A=\left\{n \in N \mid n^{2} \leq n+10,000\right\}, B=\{3 k+1 \mid k \in N\}$ and $C=\{2 k \mid k \in N\}$, then the sum of all the elements of the set $A \cap(B-C)$ is equal to $…..$

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

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

Let $\bigcup \limits_{i=1}^{50} X_{i}=\bigcup \limits_{i=1}^{n} Y_{i}=T$ where each $X_{i}$ contains $10$ elements and each $Y_{i}$ contains $5$ elements. If each element of the set $T$ is an element of exactly $20$ of sets $X_{i}$ 's and exactly $6$ of sets $Y_{i}$ 's, then $n$ is equal to