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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?

 

2)  Consider all the possible arrangements of the letters in the term "ALLSTARS." How many do not have two same letters next to each other?

 

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?

 

Thanks!

 May 11, 2019
 #1
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Why is my post flagged?

 May 11, 2019
 #2
avatar+104628 
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Mmmmm....must be a "ghost in the machine"......consider it "un-flagged"

 

cool cool cool

CPhill  May 11, 2019
 #3
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+1

Thanks! :)

Rudram592  May 11, 2019
 #4
avatar+104628 
+1

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

 

 

cool cool cool

 May 11, 2019
 #8
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Thanks!

Rudram592  May 11, 2019
 #5
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+1

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

 

 

cool cool cool

 May 11, 2019
 #6
avatar+104628 
+1

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

 

That would be  100  - 41   = 59

 

 

cool cool cool

 May 11, 2019
 #7
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The correct answer was 60. I think you just forgot to add one since the number was inclusive. 

Rudram592  May 11, 2019
 #11
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Eh...I left out 60....the other multiple of 4, 5

 

So

 

(25 + 45) - 5  =  40

 

And 

 

100 - 40  =  60  !!!

 

cool cool cool

CPhill  May 11, 2019
 #9
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xxxxxxxxxx

 May 11, 2019
edited by Guest  May 11, 2019
 #12
avatar+4325 
+1

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

 May 11, 2019
 #13
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+1

 

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. 
 

 May 11, 2019

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