1. Represent the above information in a Venn-diagram.
  2. How many students study all the subjects?
  3. How many students are there altogether?

Solution:

Let U be the set of total students.

Let E, H and S represent the sets of students learning English, History and Science, respectively.

Here,
$n(E) = 20$
$n(H) = 18$
$n(S) = 21$
$n_o(S) = 10$
$n_o(E) = 7$
$n_o(E \cap S) = 6$
$n_o(S \cap H) = 3$

We know,
$n(S) = n_o(S) + n_(E \cap S) + n_o(S \cap H) + n(E \cap H \cap S)$

$or, 21 = 10 + 6 + 3 + n(E \cap H \cap S)$

$or, 21 = 19 + n(E \cap H \cap S)$

$\therefore n(E \cap H \cap S) = 2$

So, 2 students study all the subjects.



And,
$n(E) = n_o(E) + n_o(E \cap S) + n_o(E \cap H) + n(E \cap H \cap S)$

$or, 20 = 7 + 6 + n_o(E \cap H) + 2$

$or, 20 = 15 + n_o(E \cap H) $

$\therefore n_o(E \cap H) = 5$


Now,
$n(U) = n(E) + n(S) + n(H) -2× n(E \cap H \cap S) - n_o(E \cap H) - n_o(E \cap S) - n_o(S \cap H)$

$= 20 + 18 + 21 -2×2 - 5 - 6 - 3$

$= 59 - 4 - 14$
$= 59 - 18$
$= 41$
Hence, the required total number of students surveyed is 41.

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