If the probability that the random variable $X$ takes the value $x$ is given by $P(X=x) = k(x+1)3^{-x}, x = 0, 1, 2, 3 \dots$, where $k$ is a constant, then $P(X \geq 3)$ is equal to

Let $A = {1, 2, 3, ..., 100}$ and $R$ be a relation on $A$ such that $R = {(a, b) : a = 2b + 1}$. Let $(a_1, a_2), (a_2, a_3), (a_3, a_4), ..., (a_k, a_{k+1})$ be a sequence of $k$ elements of $R$ such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k , for which such a sequence exists, is equal to :

Let $X = {1, 2, 3, 4, 5}$. The number of different ordered pairs $(Y, Z)$ that can be formed such that $Y \subseteq X$, $Z \subseteq X$ and $Y \cap Z$ is empty, is:

Let $A = {0, 1, 2, 3, 4, 5}$. Let $R$ be a relation on $A$ defined by $(x, y) \in R$ if and only if $\max{x, y} \in {3, 4}$. Then among the statements (S1): The number of elements in $R$ is $18$, and (S2): The relation $R$ is symmetric but neither reflexive nor transitive

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to ___________.