Question

2. There are 75 students in the Travel Club. They discovered that 27 members have visited Mexico,
34 have visited Canada, 12 have been to England, 18 have visited both Mexico and Canada, 6 have
been only to England, and 8 have been only to Mexico. Some club members have not been to any
of the three foreign countries and, curiously, and equal number have been to
all three countries.
a) How many students have been to all three countries?
b) How many students have been only to Canada?​

Answer 1

Answer:

Step-by-step explanation:

Total students = 75

Visited Mexico  = 27   = M

Visited Canada = 34  = C

Visited England = 12  = E

Visited Mexico & Canada = 18   = M∩C

Only to England = 6  =   E only

Only to Mexico = 8 =  M only

Visited no country = X  

Visited all country = X    M∩C∩E

Total Students = M + C + E  -  M∩C -  M∩E - E∩C  + M∩C∩E  + No country

=> 75 = 27 + 34 + 12 - 18 -  M∩E - E∩C + X + X

=>  M∩E + E∩C = 2X - 20

Only Mexico = M - M∩E  - M∩C + M∩C∩E

=> 8 = 27 -  M∩E - 18  + X

=> M∩E = X + 1

Only England = E - M∩E  - E∩C + M∩C∩E

=> 6 = 12 - X -1 - E∩C + X

=>E∩C = 5

Only Canada = C - M∩C  - E∩C + M∩C∩E

=> Only Canada = 34 - 18 - 5 + X

=> Only Canada = 11 + X

M∩E + E∩C = 2X - 20

=> M∩E + 5 = 2X - 20

=> M∩E  = 2X - 25

M∩E = X + 1

=> 2X - 25 = X + 1

=> X = 26

its not possible that 26 people visited all countries

So some data is wrong

With Some different Data Solved

With changed data  

Total students = 40

Visited Mexico  = 17   = M

Visited Canada = 28  = C

Visited England = 10  = E

Visited Mexico & Canada = 12   = M∩C

Only to England = 3  =   E only

Only  to Mexico = 4 =  M only

Visited no country = X    

Visited all country = X    M∩C∩E

Total Students = M + C + E  -  M∩C -  M∩E - E∩C  + M∩C∩E  + No country

=> 40 = 17 + 28 + 10 - 12 -  M∩E - E∩C + X + X

=>  M∩E + E∩C = 2X + 3

Only Mexico = M - M∩E  - M∩C + M∩C∩E

=> 4 = 17 -  M∩E - 12  + X

=> M∩E = X + 1

Only England = E - M∩E  - E∩C + M∩C∩E

=> 3 = 10 - X -1 - E∩C + X

=>E∩C = 6

M∩E + E∩C = 2X + 3

=> M∩E + 6 = 2X +3

=> M∩E  = 2X - 3

M∩E = X + 1

=> 2X - 3 = X + 1

=> X = 4

Only Canada = C - M∩C  - E∩C + M∩C∩E

=> Only Canada = 34 - 18 - 5 + 4

=> Only Canada = 15