Question

Out of $100$ students; $15$ passed in English, $12$ passed in Mathematics, $8$ in Science, $6$ in English and Mathematics, $7$ in Mathematics and Science, $4$ in English and Science; $4$ in all the three passed. Then
(i) The number of students passed in English and Mathematics but not in Science is
(ii) The number of students only passed in Mathematics is
(iii) The number of students passed in more than one subject is

	(i)	(ii)	(iii)
(a)	$1\,\,\,$	$3\,\,\,$	$2\,\,\,$
(b)	$2$	$3$	$9$
(c)	$3$	$2$	$9$
(d)	$2$	$9$	$3$

• A. a
• B. b
• C. c
• D. d

Answer 1

Answer: (B)

Let $U$, $E$, $M$ and $S$ be denote the total number of students, number of students passed in English, Mathematics and Science, respectively.
Here, $n(U) = 100$, $n(E) = 15$, $n(M) = 12$, $n(S) = 8$, $n(E \cap M) = 6$,
$n(M \cap S) = 7$, $n(E \cap S) = 4$ and $n (E \cap M \cap S) = 4$
(i) $\therefore $ The number of students passed in English and Mathematics but not in Science
$= n ( E \cap M \cap \bar{S})$
$= n(E \cap M) - n(E \cap M \cap S) = 6 - 4 = 2$
(ii) The number of students only passed in Mathematics
$= n(M \cap \bar{E} \cap \bar{S})$
$= n(M) - n(M \cap E) - n(M \cap S) + n(M \cap E \cap S)$
$= 1 2 -6 -7 + 4 = 16-13 = 3$
(iii) The number of students passed in more than one subject
$= n(M \cap E) + n(M \cap S ) + n(S \cap E) - 2n(M \cap E \cap S)$
$= 6 + 7 + 4 - 2(4) = 17-8 = 9$