# 16 friends were discussing movies. 9 have seen Attack of the Killer Tomatoes, 12 have seen The Revenge of Killer Cucumbers & 6 have seen both. How many have seen neither?

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Of the 16 friends discussing movies, 9 have seen Attack of the Killer Tomatoes, 12 have seen The Revenge of Killer Cucumbers & 6 have seen both.

The set of friends that have seen Attack of the Killer Tomatoes has 9 elements, the set of children that has seen The Revenge of Killer Cucumbers is 12 and the intersection of the two sets has 6 elements.

For two elements A and B, with the number of elements n(A) and n(B),

`n(AuuuB) = n(A) + n(B) - n(AnnnB)`

Here `n(A) = 9, n(B) = 12 and n(AnnnB) = 6`

The number of friends that have seen the movies is `n(AuuuB)` `n(AuuuB) = 9 + 12 - 6 = 15`

The total number of friends is 16, this gives the number of friends that has seen neither as 16 - 15 = 1.

**1 friend has seen neither of the movies.**

To me, the easiest way to solve this would be to make a Venn Diagram of which people saw which movie. Label one circle "AKT" or Attack of the Killer Tomatoes, and one circle "RKC" or Revenge of the Killer Cucumbers, with the Both section in the middle.

First, you know that there are 6 friends that saw both, so put 6 tallies in the both section. Then you know that 9 friends total have seen AKT. So, you would do [9-6] which equals 3. So three friends have seen just AKT. You would put three tally marks in the AKT circle. Next you have 12 friends total that saw Revenge of the Killer Cucumbers. So, you would do [12-6] which equals 6. So 6 friends have seen just RKC. You would put 6 tally marks in the RKC circle. If you add up all the tallies in your chart [6+6+3] you should get 15. Therefore, **1 friend has seen neither of the movies.**