Math  /  Data & Statistics

QuestionImport favorites Booking.com McAfee Security There are 37 students in the University Travel Club. The following information and Venn diagram show how many students traveled to Germany, France, and Spain. How many students have been to France or Spain but not Germany? 17 members have visited Germany 13 have been to Spain 19 have visited France 10 have been to Germany and France 6 have only been to France 5 have only been to Germany 3 have been to only Spain and France 3 have been to all three countries Some of the members have not been to any of the three

Studdy Solution

STEP 1

1. We need to find the number of students who have visited France or Spain but not Germany.
2. We will use the information provided about the number of students who visited each combination of countries.
3. We will employ the principle of inclusion-exclusion to solve the problem.

STEP 2

1. Define the sets for students who visited each country.
2. Determine the number of students in each relevant subset.
3. Calculate the number of students who have visited France or Spain but not Germany.

STEP 3

Define the sets for students who visited each country: - Let GG be the set of students who visited Germany. - Let FF be the set of students who visited France. - Let SS be the set of students who visited Spain.
Given: G=17,F=19,S=13|G| = 17, \quad |F| = 19, \quad |S| = 13 |G \cap F| = 10, \quad |F \cap S| = 3, \quad |G \cap S| = $unknown GFS=3|G \cap F \cap S| = 3 F(GS)=6,G(FS)=5|F \setminus (G \cup S)| = 6, \quad |G \setminus (F \cup S)| = 5

STEP 4

Determine the number of students in each relevant subset. First, find GS|G \cap S| using the principle of inclusion-exclusion for GFS|G \cup F \cup S|: GFS=G+F+SGFFSGS+GFS|G \cup F \cup S| = |G| + |F| + |S| - |G \cap F| - |F \cap S| - |G \cap S| + |G \cap F \cap S|

STEP 5

Since some members have not been to any country, let xx be the number of such students. Then, the total number of students is: 37x=GFS 37 - x = |G \cup F \cup S|
Substitute and solve for GS|G \cap S|: 37x=17+19+13103GS+3 37 - x = 17 + 19 + 13 - 10 - 3 - |G \cap S| + 3 37x=39GS 37 - x = 39 - |G \cap S| GS=3937+x |G \cap S| = 39 - 37 + x GS=2+x |G \cap S| = 2 + x

STEP 6

Calculate the number of students who have visited France or Spain but not Germany: (FS)G=FSFSG |(F \cup S) \setminus G| = |F \cup S| - |F \cup S \cap G|
First, find FS|F \cup S|: FS=F+SFS |F \cup S| = |F| + |S| - |F \cap S| FS=19+133=29 |F \cup S| = 19 + 13 - 3 = 29

STEP 7

Next, calculate FSG|F \cup S \cap G|: FSG=(FG)(SG) |F \cup S \cap G| = |(F \cap G) \cup (S \cap G)| Using the principle of inclusion-exclusion: (FG)(SG)=FG+SGFGS |(F \cap G) \cup (S \cap G)| = |F \cap G| + |S \cap G| - |F \cap G \cap S| (FG)(SG)=10+(2+x)3 |(F \cap G) \cup (S \cap G)| = 10 + (2 + x) - 3 (FG)(SG)=9+x |(F \cap G) \cup (S \cap G)| = 9 + x

STEP 8

Finally, calculate (FS)G|(F \cup S) \setminus G|: (FS)G=FS(FG)(SG) |(F \cup S) \setminus G| = |F \cup S| - |(F \cap G) \cup (S \cap G)| (FS)G=29(9+x) |(F \cup S) \setminus G| = 29 - (9 + x) (FS)G=20x |(F \cup S) \setminus G| = 20 - x
Given the total number of students who have not been to any country is an unknown xx, and assuming x=0x = 0 (no students have been to none of the countries), we get: (FS)G=20 |(F \cup S) \setminus G| = 20
Thus, the number of students who have been to France or Spain but not Germany is 2020.

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