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

The number of elements in the set $\left\{ n \in N : 10 \leq n \leq 100\right.$ and $3^{ n }-3$ is a multiple of $7\}$ is $........$.

Let $S=\{4,6,9\}$ and $T=\{9,10,11, \ldots, 1000\}$. If

$A=\left\{a_{1}+a_{2}+\ldots+a_{k}: k \in N, a_{1}, a_{2}, a_{3}, \ldots, a_{k} \in S\right\}$ then the sum of all the elements in the set $T - A$ is equal to $......$

If $A = \{x, y\}$ then the power set of $A$ is

The number of elements in the set $\left\{n \in Z :\left|n^2-10 n+19\right| < 6\right\}$ is $...........$

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 $.....$