According to a student survey, 16 students liked history, 19 liked english, 18 liked math, 8 liked math and english, 5 liked history and english, 7 liked history and math, 3 liked all three subjects. draw a venn diagram to answer the following: a) how many students were in the survey? b) how many students liked only math? c) how many students liked english and math, but not history.

When building a Venn Diagram, I always start from the area with the most overlap to the areas of least overlap. Once you have placed the 3 in the middle, you have counted those people, and therefore you must subtract them from the other surveys. Example: since there are 3 people that like all three subjects, now only have 5 students that like just math and English instead of 8. 

Therefore:
A) 36 Students were in the survey
*Add all the numbers within the Venn diagram up. Overlapping doesn't matter because no one is double counted. 

B) 6 People liked only Math
*Can't touch any other circle but Math

C) 20 Students liked English and math, but not history
*You add 9+5+6, since these bubbles are not overlapping with history. 

I Hope this helps and let me know if you have any further questions!