There are a total of 141 foreign language students in a high school where they offer Spanish, French, and German. There are 15 students who take French and German
24 students who take French and Spanish
6 students who take German and Spanish
5 students who take all three languages.
If there are 67 Spanish students, 66 French students, and 58 German students, find:

a. How many students take French only.
b. How many students take French AND German but not Spanish.
c. How many students take French OR German but not Spanish.

a) 32 students

b) 10 students

c) 84students

Step-by-step explanation:

Let the total number of student be n(U) = 141

Those  who take French and German = n(F∩G) = 15

Those who take French and Spanish = n(F∩S) = 24

Those who take German and Spanish = n(G∩S) = 6

Those who take all the three subjects = n(F∩G∩S) = 5

Spanish student n(S) = 67

French students n(F)= 66

German students n(G) = 58

Find the venn diagram attached for clarification

a) From the diagram, number of student that took french only = x = n(G'∪S'∪F)

n(G'∪S'∪F) = n(F) - (10+5+19)

n(G'∪S'∪F) = 66 - 34

n(G'∪S'∪F)  = 32

Hence 32 students took only French

b) Number of students that take French AND German but not Spanish is represented using the set notation

n[(F∩G)∪S']

This is the intersection of F and G excluding Spanish. From the diagram, the value is 10. Hence;

n[(F∩G)∪S'] = 10

Hence 10 students took French AND German but not Spanish.

c)  Before we can get the number of students that take French OR German but not Spanish, we need to get those that take German only first;

y = n(G∪S'∪F') = n(G) - (10+5+1)

n(G∪S'∪F') = 58 - 16

n(G∪S'∪F') = 42 students

Number of students take French OR German but not Spanish = x + y + 10

Number of students take French OR German but not Spanish = 32 + 42 + 10 = 84 (Note that students that studied French only and German only are inclusive compared to b above that they are excluded)

Hence 84 students took French OR German but not Spanish.

hello from mrbilldoesmath!

answer:   (7x + 6y) ( 49x^2 - 42xy + 36y^2)

discussion:

in general, the factorization of a^3 + b^3 is

a^3 + b^ 3 = (a + b) ( a^2 - ab + b^2)

in our case note that 343 = 7^3 and 216 = 6^3 so the equation becomes:

343x^3 + 216y^3 = (7x)^3 + (6y)^3

if you think of a (above) as 7x and b as 6y the factorization becomes

343x^3 + 216y^3 =

((7x + 6y) (   (7x)^2 - 7x*6y + (6y)^2)   =

(7x + 6y) ( 49x^2 - 42xy + 36y^2)

you,

mrb

(a + b) (a^2 - a b + b^2)