In a survey of $400$ students in a school,$100$ were listed as taking apple juice,$150$ as taking orange juice and $75$ were listed as taking both apple as well as orange juice. Find how many students were taking neither apple juice nor orange juice.

The number of integers from $1$ to $1000$ which are divisible by $2$ or $3$ is

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

The sum of all the elements of the set $\{\alpha \in \{1, 2, \ldots, 100\} : \operatorname{HCF}(\alpha, 24) = 1\}$ is

The probabilities of three events $A, B$ and $C$ are given by $P(A)=0.6, P(B)=0.4$ and $P(C)=0.5$. If $P(A \cup B)=0.8, P(A \cap C)=0.3, P(A \cap B \cap C)=0.2, P(B \cap C)=\beta$ and $P(A \cup B \cup C)=\alpha$,where $0.85 \leq \alpha \leq 0.95$,then $\beta$ lies in the interval:

In a cinema hall,the charge per person is $₹ 200$. On the first day,only $60 \%$ of the seats were filled. The owner decided to reduce the price by $20 \%$ and there was an increase of $50 \%$ in the number of spectators on the next day. The percentage increase in the revenue on the second day was (in $\%$)