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$n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)$

In this question, we need to determine the total number of students who do not have any of these hobbies of rock climbing as well as sky watching, who have the hobby of rock climbing only and who have the hobby of sky watching only for which we need to evaluate the intersection of both the hobbies.

Let $n\left( A \right)$ = The number of students whose hobby is rock climbing.

$n\left( B \right) = $ The number of students whose hobby is sky watching.

$n\left( {A \cap B} \right) = $ The number of students follows both hobbies.

We have,

$n\left( {A \cup B} \right) = $ Number of students whose hobby is either rock climbing or sky watching

$

n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) \\

= 130 + 180 - 110 \\

= 200 \\

$

Here, the total number of students = 220

So, the number of students who do not have any of these two hobbies is$220 - 200 = 20$

Also, the number of students who follow the hobby of rock climbing only is given as:

$

n\left( {A - B} \right) = n\left( A \right) - n\left( {A \cap B} \right) \\

= 130 - 110 \\

= 20 \\

$

Again, the number of students who follow the hobby of sky watching only is given as:

$

n\left( {B - A} \right) = n\left( B \right) - n\left( {A \cap B} \right) \\

= 180 - 110 \\

= 70 \\

$

Hence,

The numbers of students who do not have any of these hobbies are 20.

The number of students who have the hobby of rock climbing are 20.

The numbers of students who have the hobby of sky watching are 70.