Examples on Principle of inclusion and Exclusion

In this class, We discuss Examples on Principle of inclusion and Exclusion.

The reader should have prior knowledge of inclusion and exclusion formulae. Click Here.

Example 1:

A total of 1232 students take courses in Spanish.

879 students have taken courses in French.

114 taken courses in Russian.

100 students take courses in Spanish and French.

25 students have taken courses in Spanish and Russian

12 students have taken courses in French and Russian.

A total of 2092 students have taken courses in at least Spanish, French, or Russian.

How many students have taken courses in all the three languages?

Solution:

|S| = 1232

|F| = 879

|R| = 114

|S ∩ F| = 100

|S ∩ R| = 25

|F ∩ R| = 12

|S ∪ F ∪ R| = 2092

We know the principle of Inclusion and exclusion formulae.

|A ∪ B ∪ C| = |A| + |B| + |C| – |A ∩ B| – |A ∩ C| – |B ∩ C| + |A ∩ B ∩ C|

2092 = 1232 + 879 + 114 – 100 – 25 -12 + |S∩F∩R|

|S∩F∩R| = 4

Example 2:

In a class of 50 students.

20 students play football

16 students play hockey

10 students play both.

Find the number of students who play neither.

Solution:

|F| = 20

|H| = 16

|F ∩ H| = 10

|F ∪ H| = |F| + |H| – |F ∩ H|

|F ∪ H| = 20 + 16 – 10

= 26

Who play neither = 50 – 26

= 24