Question

at a high school science fair, 34 students received awards for scientific project.14 awards were given for projects in biology,13 in chemistry,21 in physics, and 3 students received awards in all 3 subjects areas

1.how many received awards for exactly two subject areas?

2.how many received awards for exactly one subject area?

Answer 1

Consider the Venn diagram above..In light of the given question we can say ,

      n(A U B U C)  =  n(A) U n(B) U n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C)

==>  n(A ∩ B) +  n(B ∩ C) + n(A ∩ C)  =  n(A) U n(B) U n(C)  -   n(A U B U C) + n(A ∩ B ∩ C)

==>  n(A ∩ B) +  n(B ∩ C) + n(A ∩ C)  =  48 - 34 + 3

==>   n(A ∩ B) +  n(B ∩ C) + n(A ∩ C)  =  17

==>   y + s + p + 3r  =  17 [According Venn diagram]

==>   y + s + p   =   17  - 3 * 3   = 8

==> n(exactly 2 awards received)  =  8  ...............(1)

Now again using Venn diagram we have :

          x + z + t + y + s + p + r  = 34 (n(A U B U C)

==>    x + z + t  + 8 + 3  =  34

==>    x + z  + t    =   34 - 11

==>  n(exactly 1 award received)   =  23