In a class of $35$ students, $24$ like to play cricket and $16$ like to play football. Also, each student likes to play at least one of the two games. How many students like to play both cricket and football?
Let $X$ be the set of students who like to play cricket and $Y$ be the set of students who like to play football. Then $X \cup Y$ is the set of students who like to play at least one game, and $X \cap Y$ is the set of students who like to play both games.
Given $\quad n( X )=24, n( Y )=16, n( X \cup Y )=35, n( X \cap Y )=?$
Using the formula $n( X \cup Y )=n( X )+n( Y )-n( X \cap Y ),$ we get
$35=24+16-n( X \cap Y )$
Thus, $n( X \cap Y )=5$
i.e., $\quad 5$ students like to play both games.
In a town of $10,000$ families it was found that $40\%$ family buy newspaper $A, 20\%$ buy newspaper $B$ and $10\%$ families buy newspaper $C, 5\%$ families buy $A$ and $B, 3\%$ buy $B$ and $C$ and $4\%$ buy $A$ and $C$. If $2\%$ families buy all the three newspapers, then number of families which buy $A$ only is
A survey shows that $73 \%$ of the persons working in an office like coffee, whereas $65 \%$ like tea. If $x$ denotes the percentage of them, who like both coffee and tea, then $x$ cannot be
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.
In a class of $140$ students numbered $1$ to $140$, all even numbered students opted Mathematics course, those whose number is divisible by $3$ opted Physics course and those whose number is divisible by $5$ opted Chemistry course. Then the number of students who did not opt for any of the three courses is
Let $\mathrm{U}$ be the set of all triangles in a plane. If $\mathrm{A}$ is the set of all triangles with at least one angle different from $60^{\circ},$ what is $\mathrm{A} ^{\prime} ?$