The 38 movies in the video store fall into the following three

Question

The 38 movies in the video store fall into the following three categories: 10 action; 20 drama; 18 comedy
5 are both action and drama; 3 are both action and comedy; 4 are both drama and comedy
How many action-drama-comedies are there?

38=10+20+18-5-3-4
38=36
38-36=ADC

Answer -> 2

A. 1
B. 2
C. 3
D. 4
E. 5

Solution:
I don't like drawing Venn diagrams because it is time consuming. I tend to use the formula:

(Total in Group 1) + (Total in Group 2) + (Total in Group 3) – (Overlap of 1 and 2) – (Overlap of 1 and 3) – (Overlap of 2 and 3) –   + (total in none of the groups) = Overall total

10 + 20 + 18 - 5 - 3 - 4 - 2(x) + 0 = 38.
=> 48 - 12 - 2x = 38
=> 36 - 2x = 38
=> x=-1

But the answer is:
 X = 2

Where am I going wrong? I used the method suggested by Bunuel in this thread:
gmatclub dot com/forum/at-a-certain-resort-each-of-the-39-food-service-employees-133276.html

Bunuel · Accepted Answer

The 38 movies in the video store fall into the following three categories: 10 action, 20 drama, and 18 comedy. However, some movies are classified under more than one category: 5 are both action and drama, 3 are both action and comedy, and 4 are both drama and comedy. How many action-drama-comedies are there?

A. 1
B. 2
C. 3
D. 4
E. 5

Show Spoiler

Solution:
I don't like drawing Venn diagrams because it is time consuming. I tend to use the formula:

(Total in Group 1) + (Total in Group 2) + (Total in Group 3) – (Overlap of 1 and 2) – (Overlap of 1 and 3) – (Overlap of 2 and 3) – [2 * (Overlap of 1, 2, and 3)] + (total in none of the groups) = Overall total

10 + 20 + 18 - 5 - 3 - 4 - 2(x) + 0 = 38.
=> 48 - 12 - 2x = 38
=> 36 - 2x = 38
=> x=-1

But the answer is:
X = 2

Where am I going wrong? I used the method suggested by Bunuel in this thread:
gmatclub dot com/forum/at-a-certain-resort-each-of-the-39-food-service-employees-133276.html

Show Answer

Official Answer

B

Narenn · Answer

Hi Saint, Check if this works for you, quite basic method though Given........ Total = 38 A=10 D=20 C=18 AD=5 AC=3 DC=4 ALL=x (Assume action-drama-comedies be x ) At first we have to remove the movies that belong to all (i.e.action-drama-comedy) categories from every subset. Total = 38 A=10 - x D=20 - x C=18 - x AD only =5 - x AC only =3 - x DC only =4 - x ALL=x At second step we have to remove movies that belong to only two categories (AD only, AC only, DC only) from the main subsets (i.e A, D, C) So we have that, A only =(10 - x) - (5-x) - (3-x) D only =(20 - x) - (5-x) - (4-x) C only =(18 - x) - (3-x) - (4-x) AD only =5 - x AC only =3 - x DC only =4 - x ALL=x So finally we have that, Total = A only + D only + C only + AD only + AC only + DC only + ALL 38 = 2 + x + 11 + x + 11 + x + 5 - x + 3 - x + 4 - x + x 38 = 36 + x x = 2 This method is just the algebraical version of venn diagram. IMO solving thru venn diagram would be much easier and faster though. Hope that helps!

TirthankarP · Answer

I agree that drawing venn diagram is a time taking thing. But you should be able to picturize the venn diagram in your mind to correctly understand the problem

As shown in the venn diagram below: Venn_Diagram.png a = only action b = only drama c= only comedy d = both action and drama but not comedy e = both action and comedy but not drama f = both drama and comedy but not action g = all three As per the question, Both action and drama (including comedy) = d + g = 5 Both action and comedy (including drama) = e + g = 3 Both drama and comedy (including action) = f + g = 4 Action = a + d + g + e = 10 Drama = b + d + f + g = 20 Comedy = c + e + f + g = 18 As per your formula: (Total in Action) + (Total in Drama) + (Total in Comedy) – (Overlap of A and D) – (Overlap of D and C) – (Overlap of A and C) – + (total in none of the groups) = Overall total => (a + d + g + e) + (b + d + f + g ) + (c + e + f + g ) – (d + g ) – (e + g ) – (f + g ) – (2* g ) + 0 = 38 Here you can see that you are adding g thrice but subtraction g 5 times (3 + 2) So instead of subtracting 2g, you have to add “g” once as you have already added g thrice and subtracted thrice earlier Hence the correct formula should be: (a + d + g + e) + (b + d + f + g ) + (c + e + f + g ) – (d + g ) – (e + g ) – (f + g ) + ( g ) + 0 = 38 => 10 + 20 + 18 - 5 - 3 – 4 + g = 38 => 48 – 12 + g = 38 => 36 + g = 38 => g = 38 – 36 = 2

daniformic · Answer

The 38 movies in the video store fall into the following three categories: 10 action; 20 drama; 18 comedy 
5 are both action and drama; 3 are both action and comedy; 4 are both drama and comedy
How many action-drama-comedies are there?

38=10+20+18-5-3-4
38=36
38-36=ADC

Answer -> 2

Rebaz · Answer

Bunuel   KarishmaB

What about the well known formula for the union of all three.

Let X be the number of movies in all three section (the inner-most)

10+20+18-(12+2X)=38 the 12 is the sum of the movies that are in two catogories(5+4+3=12)

10+20+18-(12+2X)=38

48-12-2X=38

36-2X=38

2X=-2

X=-1, which does not make sense.

Where am i going wrong? How is it possible that most often to find the inner most is the above approach correct but not for this question.

Bunuel · Answer

There are two formulae and you are using the wrong one. Check here: https://gmatclub.com/forum/the-38-movie ... 60697.html Hope it helps.

JT2916 · Answer

brunel  
If I use the first formula:

x = group overlaps/all three

Total=A+B+C−(sum of 2−group overlaps)+(all three)+Neither
=> 38 = 10 + 18 + 20 - (5 + 4 + 3 - (x)) + x
=> 2 = 2x
=> x = 1

What is wrong in this?

Bunuel · Answer

You don't need to subtract x. Please study this: https://gmatclub.com/forum/advanced-ove ... 44260.html

voluptatemfuga · Answer

I am confused regarding the formula...
on other questions you have mentioned this formula as T = A+B+C - (in exactly 2) - 2*(in exactly 3) 
but here you have mentioned the formula as T = A+B+C - (in exactly 2) + (in exactly 3)
my doubt is when to use what formula

Bunuel · Answer

Explained here ADVANCED OVERLAPPING SETS PROBLEMS: https://gmatclub.com/forum/advanced-ove ... 44260.html

The 38 movies in the video store fall into the following three

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

The 38 movies in the video store fall into the following three

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards