Geometry
posted by John on .
Sandy has a long list of 100 suitors. She decides to label them according to if they are tall, dark or handsome. She finds that each of her suitors fall into at least one of the categories, and 65 of them are tall, 76 of them are dark and 53 of them are handsome. If 20 of the suitors are in exactly 2 of the categories, how many of her suitors are tall, dark and handsome?

I made a Venn diagram with 3 intersecting circles T, D, and H
I labeled the intersection of all 3 as x, our unknown.
I labeled the part consisting only of T and D (no x) as a
I labeled the part consisting only of T and H as b
I labeled the part consisting only of D and H as c
so a+b+c = 20 , according to my interpretation
65 + 76bx + 53axc = 100
194  (a+b+c)  2x = 100
194  20  2x = 100
2x = 74
x = 37
so 37 are tall , dark and hansome