In a survey of 2000 Indian tourists who arrived in Nepal, 65% wished to visit Pashupati, 50% wished to visited Chandragiri and 45% wishod to visit Manakamana. Similarly, 35% wished to visit Pashupati and Chandragiri, 25% to Chandragiri and Manakamana and 20% to Manakamana and Pashupati. If 5% wished to visit none of these places, find the number of tourists who wished to visit all these three places. Also, show the above data in a Venn-diagram.


Answer: 300

Solution:

Let the set of total number of tourists surveyed be represented by U.
Let the set of people who wished to visit Pashupati, Chandragiri, and Manakama be represented by P, C and M, respectively.

According to the question,
$n(U) = 2000$
$n(P) = 65%$
$n(C) = 50%$
$n(M) = 45%$
$n(P \cap C) = 35%$
$n(C \cap M) = 25%$
$n(P \cap M) = 20%$
$n(\overline{P \cup C \cup M}) = 5%$
To find: $n(P \cap C \cap M) = ?$

Let $n(P \cap C \cap M) = x$

Using formula, we have,

$ n(U) = n(P) + n(C) + n(M) - n(P \cap C) - n(C \cap M) - n(P \cap M) + n(P \cap C \cap M) + n(\overline{P \cup C \cup M})$

$or, 2000 = 65% + 50% + 45% - 35% - 25% - 20% + x% + 5%$

$or, 2000 = x + 85%$

$or, x = 2000 - 80%$

$or, x = 2000 - 80/100 × 2000$

$or, x = 2000 - 1700$

$\therefore x = 300$

Hence, 300 tourists wished to visit all the three places.