In a survey of 900 tourists who arrived in Nepal during Visit Nepal 2020, 450 preferred to go trekking, 300 preferred rafting, 400 preferred forest safari and 100 preferred none of these activities. If 200 preferred trekking and rafting, 110 preferred trekking and safari, 100 preferred rafting and safari,

1. How many of them preferred all of these activities?
2. Show the above information in a Venn-diagram.

Solution:

Let U be the set of total tourists who were surveyed.

Let T, R, and F represent the sets of people who preferred Trekking, Rafting, and Forest Safari, respectively.

Given,

$n(U) = 900$

$n(T) = 450$

$n(R) = 300$

$n(F) = 400$

$n(T \cup R \cup F)^c = 100$

$n(T \cap R) = 200$

$n(T \cap F) = 110$

$n(R \cap F) = 100$

To find:$ n(T \cap \R \cap F) = ?$

Using formula,

$n(T \cup R \cup F) = n(U) - n(T \cup R \cup F)^c$

$= 900 - 100$

$= 800$

Now,

$n(T \cup R \cup F) = n(T) + n(R) + n(F) - n(T \cap R) - n(T \cap F) - n(R \cap F) + n(T \cap R \cap F)$

$or, 800 = 450 + 300 + 400 - 200 - 110 - 100 + n(T \cap R \cap F)$

$or, 800 = 1150 - 410 + n(T \cap R \cap F)$

$or, 800 = 740 + n(T \cap R \cap F)$

$or, n(T \cap R \cap F) = 800 - 740$

$\therefore n(T \cap R \cap F) = 60$

Hence, 60 tourists preferred all of these activities.

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