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.
A class has $175$ students. The following data shows the number of students obtaining one or more subjects. Mathematics $100$, Physics $70$, Chemistry $40$; Mathematics and Physics $30$, Mathematics and Chemistry $28$, Physics and Chemistry $23$; Mathematics, Physics and Chemistry $18$. How many students have offered Mathematics alone
A group of $40$ students appeared in an examination of $3$ subjects - Mathematics, Physics Chemistry. It was found that all students passed in at least one of the subjects, $20$ students passed in Mathematics, $25$ students passed in Physics, $16$ students passed in Chemistry, at most $11$ students passed in both Mathematics and Physics, at most $15$ students passed in both Physics and Chemistry, at most $15$ students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is___________.
Out of $800$ boys in a school, $224$ played cricket, $240$ played hockey and $336$ played basketball. Of the total, $64$ played both basketball and hockey; $80$ played cricket and basketball and $40$ played cricket and hockey; $24$ played all the three games. The number of boys who did not play any game is
In a class of $100$ students, $55$ students have passed in Mathematics and $67$ students have passed in Physics. Then the number of students who have passed in Physics only is
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