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

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 $A =\{ x \in R :| x +1|<2\}$ and $B=\{x \in R:|x-1| \geq 2\}$. Then which one of the following statements is NOT true ?

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

$2n (A / B) = n (B / A)$ and $5n (A \cap B) = n (A) + 3n (B) $, where $P/Q = P \cap Q^C$ . If $n (A \cup B) \leq 10$ , then the value of $\frac{{n\ (A).n\ (B).n\ (A\  \cap\  B)}}{8}$ is 

If $X = \{ {8^n} - 7n - 1:n \in N\} $ and $Y = \{ 49(n - 1):n \in N\} ,$ then