set theory

[bmwhype]

A position in the UN has 125 applicants. Of these applicants, 70 speak French, 65 Spanish, 65 German, 45 speak French and Spanish, 35 speak Spanish and German, 40 speak French and German, and 35 speak all three languages. How many applicants speak none of the languages?


Can someone solve this with Set Theory approach
Is this formula correct?

A only + B only + C only - Both - 2*(ALL 3) + None
where Both = AB + BC + AC + ABC

[krusta80]

A position in the UN has 125 applicants. Of these applicants, 70 speak French, 65 Spanish, 65 German, 45 speak French and Spanish, 35 speak Spanish and German, 40 speak French and German, and 35 speak all three languages. How many applicants speak none of the languages?

ALL = (Exactly 0) + (Exactly 1) + (Exactly 2) + (Exactly 3)

ALL = 125
Exactly 3 = 35
Exactly 2 = (At least 2) - 3*(Exactly 3) = 45+35+40 - 105 = 15
Exactly 1 = (At least 1) - 2*(At least 2) + 3*(Exactly 3) = 200 - 240 + 105 = 65
Exactly 0 = ALL - (Exactly 1) - (Exactly 2) - (Exactly 3) = 125 - 35 - 15 - 65 = 10

[DWarrior]

Your formula is correct, but it won't work for this problem. We're not given French-only, Spanish-only, German-only, we're given ANY French, ANY Spanish, ANY German, so FSG is part of French, Spanish, and German.

The formula should be:
(A+B+C) - (AB+AC+BC) + ABC

70+65+65-45-35-40+35=115

total = 125, so 10 don't speak any

[krusta80]

For $hits and giggles:

Given n categories, we have the following to solve for the number of entities assigned to exactly i categories.
(Number of entities assigned to exactly i categories) = (Number assigned to at least i categories) - Sum((n-i)Cj*nCi*(Number of entities assigned to exactly i+j categories)), with j from 1 through n-i.
E(i) = A(i) - Sum((n-i)Cj*nCi*E(i+j))
E(0) = Total entities - Sum(E(i)), where i = 1 through n

[bmwhype]

Originally Posted by DWarrior

Your formula is correct, but it won't work for this problem. We're not given French-only, Spanish-only, German-only, we're given ANY French, ANY Spanish, ANY German, so FSG is part of French, Spanish, and German.

The formula should be:
(A+B+C) - (AB+AC+BC) + ABC

70+65+65-45-35-40+35=115

total = 125, so 10 don't speak any

thank you

i could not figure out why some questions that approach did not work until now

[eltonr]

DWarrior,

Since you said we are not given F only or S only or G only, we cannot use the formula listed by bmwhype.

Now here is a problem which was on the forum a few days ago...
I solved it using the formula and I was told that the soln was correct.
But here too it is not said that they buy milk only or cheese only or butter only.

So is the soln wrong ???
How do we distinguish the two cases ???

Thanks
Elton

Tricky PS question that may involve a Venn Diagram
Foodmart customers regularly buy at least one of the following products: milk, chicken, or apples. 60% of shoppers buy milk, 50% buy chicken, and 35% buy apples. If 10% of the customers buy all 3 products, what percentage of Foodmart customers purchase 2 of the above products?

tamg08View Public Profile

)



It is 25%
100 = 60 + 50 + 35 - X - 2* 10
Hence X = 25

Elton

[gmatcraze]

Originally Posted by DWarrior

Your formula is correct, but it won't work for this problem. We're not given French-only, Spanish-only, German-only, we're given ANY French, ANY Spanish, ANY German, so FSG is part of French, Spanish, and German.

The formula should be:
(A+B+C) - (AB+AC+BC) + ABC

70+65+65-45-35-40+35=115

total = 125, so 10 don't speak any

I am confused with this explanation .... isn't the formula that is being used here:
n (A U B U C) = n(A) + n(B) + n(C) -n(A∩B) –n(B∩C) – n(C∩A) + n(A∩B∩C).
125 = 70+65+65-45-35-40+35
125 = 115

but the question is asking to find "none"? If I do 125 -115, I get 10 which si the required answer ... but am just wondering how none becomes part of the above formula???? Please help to clarify .... thanks

[gunestopcu]

The formula is
ABC (UNION)= A + B + C - AB - AC - BC + ABC

= 70 + 65 + 65 - 45 - 35 - 40 + 35
= 115

125 - 115 = 10 speak none of the languages.

[Ursus]

Please find the Venn Diagram attached, I found it very easy with the Venn Diagram, First enter value of All, then intersection between two, then find only 1, add(115) subtract from total.

[bose]

i got 10