Question

In a survey of 100 students, the number of students studying the various languages were found to be : English only 18, English but not Hindi 23, English and Sanskrit 8, English 26, Sanskrit 48, Sanskrit and Hindi 8, no language 24. Find:
(i) How many students were studying Hindi?
(ii) How many students were studying English and Hindi?

Answer 1

Let E, H and S be the sets of students who study English, Hindi and Sanskrit, respectively.
Also, let U be the universal set.
Now, we have:
n(E) = 26, n(S) = 48, n(E$\cap$S) = 8 and n(S$\cap$H) = 8
Also,
n(E$\cap$H$\text{'}$) = 23
$\Rightarrow$n(E) $-$ n(E$\cap$H) = 23
$\Rightarrow$26 $-$ n(E$\cap$H) = 23
$\Rightarrow$n(E$\cap$H) = 3
Therefore, the number of students studying English and Hindi is 3

n(E$\cap$H$\text{'}$$\cap$S$\text{'}$) = 18
$\Rightarrow$n(E) $-$ n{E$\cap$(H$\cup$S)'} = 18
$\Rightarrow$26 $-$ n{(E$\cap$H)$\cup$(E$\cap$S)}= 18
$\Rightarrow$26 $-$ {3 + 8 $-$ n(E$\cap$H$\cap$S)} = 18
$\Rightarrow$n(E$\cap$H$\cap$S) = 3

Also,
n(E$\text{'}$$\cap$H$\text{'}$$\cap$S$\text{'}$) = 24
$\Rightarrow$n(U) $-$ n(E$\cup$H$\cup$S) = 24
$\Rightarrow$n(E$\cup$H$\cup$S) = 76

∴ Number of students studying Hindi = n(E$\cup$H$\cup$S) $-$ n(E) $-$ n(S) + n(E$\cap$H) + n(E$\cap$S) + n(S$\cap$H) $-$ n(E$\cap$H$\cap$S)
= 76 $-$ 24 $-$ 48 + 3 + 8 + 8 $-$ 3
= 18