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1) In a survey of 100  students who watch television, 21 watch American Idol, 39 watch Lost, and 8 watch both. How many of the students surveyed watch at least one of the two shows?

(21 - 8)  =  13  = Those that watch "American Idol" only

(39 - 8)  =  31 =  Those that watch "Lost" only

So....the number that watch at least one  is  

13 + 8 +  31  =

52

3)  If all multiples of 4 and all multiples of 5 are removed from the set of integers from 1 through 100, how many integers remain?

We have 25  multiples  of 4   and  20 multiples of 5

However....we've  "double-counted"  the multiples of 4 and 5   =  20, 40, 80  and 100

So

[ 25 + 20 ] -  4   =     41

Oops...sorry....you wanted the ones that remain  !!!

That would be  100  - 41   = 59

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Hint for number 2: Count the number of possible arrangements...and subtract by the number of arrangements that the letters are next to each other. For this, try to use a superblock for (A), (L), (S)...because they all repeat twice

We make a Venn diagram. There are 21 students who watch American Idol but not Lost and 39-8=31 students who watch Lost but not American Idol. In total, there are \(13+8+31=\boxed{52} \)

 students who watch at least one of the shows.