Q 4: Find the Number of Mathematics students in a university taking at-least one of the language Mandarin, English and Japan Give the following data: 65 study Mandarin 45 study English 42 study Japanese 20 Study Mandarin and English 25 Study Mandarin an Japanese 8 study all three languages

Let's denote M for mandarin learning students, E for english learning student, J for japanese learning students. Then we have:

$|M|=65$

$|E|=45$

$|J|=42$

$|M\cap J|=25$

$|M\cap E|=20$

$|M\cap E\cap J|=8$

We need to find $|M\cup E \cup J|$

$|M\cup E \cup J|=$

$=|M|+|E|+|J|-|M\cap J|-|M\cap E|-|E\cap J|+|M\cap E\cap J|=$

$=65+45+42-25-20+8-|E\cap J|=115-|E\cap J|$

Since the information how many students study English and Japanese is missing, let's denote $EJ$ the number of students studying English and Japanese $|E\cap J|$ .

Thus number of students in a university taking at-least one of the language Mandarin, English and Japan is $115-EJ$

Answer: 115 – EJ