For billy’s birthday party his mother has baked 80 cookies. 40 cookies contain vanilla and 45 contains nuts, and 15 contain both. one of billy’s friends, jim, is allergic to both nuts and vanilla, but he loves cookies! what is the largest number of cookies that jim can eat on billy’s birthday? (note that it is fine for jim to smell nuts and vanilla as long as he is not eating anything that contains them.)

In this item, we are to determine the number of cookies that contains only vanilla or nuts. To do so, we have to subtract from the number of cookies containing vanilla or nuts, the number of cookies that contains both. 
                 
                 cookies with vanilla only = 40 - 15 = 25
                  cookies with nuts only = 45 - 15 = 30

cookies with either vanilla or nuts or both = 25 + 30 + 15 = 70

cookies without vanilla or nuts = 80 - 70 = 10

Therefore, the greatest number of cookies that Jim can eat is 10. 

The answer is 1532.75