Question

In a survey conducted on students studying in grade 10 to determine suitable place for educational trips among pokhara, lumbini, and ilam, it was found that 40 students considered pokhara to be suitable, 30 students considered lumbini to be suitable, and 45 students considered ilam to be suitable. While 15 students of that grade said that all three places are suitable, 5 students didn't express any opinion.
(a). If P, L, I denoted the set of students who prefer pokhara, lumbini, and ilam respectively, write the cardinality notation f students for whom all places are suitable.
(b). Show the above information in venn diagram.

Answer 1

Answer: b

Step-by-step explanation:

Answer 2

Answer: Sure, here's the explanation with symbols replaced by normal letters:

(a) Let's denote the sets as follows:

P: Set of students who prefer Pokhara

L: Set of students who prefer Lumbini

I: Set of students who prefer Ilam

Given:

Number of students who prefer Pokhara ∣P∣) = 40

Number of students who prefer Lumbini (∣L∣) = 30

Number of students who prefer Ilam (∣I∣) = 45

Number of students who prefer all three places (∣P∩L∩I∣) = 15

Total number of students expressing an opinion (excluding those who didn't express any opinion) (∣P∪L∪I∣) = 40+30+45−15=100

Number of students who didn't express any opinion = 5

Now, to find the number of students for whom all places are suitable, we can use the principle of inclusion-exclusion:

∣P∩L∩I∣=∣P∣+∣L∣+∣I∣−∣P∪L∪I∣+∣P∩L∩I∣

15=40+30+45−100+∣P∩L∩I∣

∣P∩L∩I∣=15+100−(40+30+45)

∣P∩L∩I∣=15+100−115

∣P∩L∩I∣=0

So, there are 0 students for whom all places are suitable.

(b) Here's the Venn diagram representing the given information:

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            _________

           /         \

          /     I     \

         /_____________\

         \      |      /

          \_____|____/

                |

         _______|______

        /               \

       /       P         \

      /___________________\

       \         |        /

        \________|______/

                |

         _______|______

        /               \

       /       L         \

      /___________________\

       \               /

        \_____________/

In the Venn diagram:

The circle labeled "P" represents the set of students who prefer Pokhara.

The circle labeled "L" represents the set of students who prefer Lumbini.

The circle labeled "I" represents the set of students who prefer Ilam.

The overlapping regions represent the students who prefer multiple places. Since there's no overlap between all three sets,

∣P∩L∩I∣=0.