At Bison High School, there are 16 students in English Club, 16 students in Science Club and 20 students in Math Club. Of these students, there are 5 students in both the English and Science Clubs; 6 students in both the Science and Math Clubs; and 8 in both the English and Math Clubs. If only 2 students are in all three clubs, how many students are in at least one of the clubs?

It would be easy if you draw a venn diagram. Anyway: There are 8 students in both Eng and Math. There are 6 students in both Math and Sci. There are 5 students in both Eng and Sci. There are 2 in all the clubs. Using the Venn diagram, you can determine the number of students in one club only. There is 1 student in Eng only. There are 4 in Math only. There are 3 in Sci only. The total would be 29 students in at least one of the clubs.

if we use Inclusion- Exclusion principle:

i show intersection symbol by "n'

E u S u M =E+ S+M-(E n S) -(E n M) - ((S n M) + (E u S u M)

E n S=5 , S n M=6 , E n M=8

E n S n M =2

so: E u S u M = 16+16+20-(5+6+8)+2

at least in one clube equals not(in no group)

all students: M = 52 at least in one group: E u S u M =35 52-35=17

if U is your universal set: |P u (not P)| = |P| + |(not P)|=|U|

note that P n (not P) = {}

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