X denotes the number of times heads that occur in $n$ tosses of a fair coin. If $P(X=4), P(X=5)$ and $P(X=6)$ ate in arithmetic progression. The largest value of $n$ is

The probability distribution of a random variable $X$ is as follows. Then, the mean of $x$ is

Two students appeared simultaneously for an entrance exam. If the probability that the first student gets qualified in the exam is $\frac{1}{4}$ and the probability that the second student gets qualified in the same exam is $\frac{2}{5}$, then the probability that atleast one of them gets qualified in that exam is

For three events $A, B$ and $C$ of a sample space, $P$ (exactly one of $A$ or $B$ occurs ) $=P$ (exactly one of $B$ or $C$ occurs) $=P($ exactly one of $C$ or $A$ occurs $)=\frac{1}{4}$. If probability of all the three events occurring simultaneously is $\frac{1}{16}$, then the probability that atleast one of the events occur is