According to a survey of 100 students, there are 40 students studying English, 30 studying French, and 25 studying Spanish. Inaddition, 8 students are studying English and French, 6 are studying English and Spanish, 5 are studying French and Spanish, and 22 are not studying any of the three languages. Which of the following is the number of students studying all three languages?
I would love to know is there any particular method to solve this kind of math?
Thanks alot!
Consider the sets
$E$ : students studying English
$F$ : students studying French
$S$ : students studying Spanish
For this kind of question you can use the formula:
$$ n(E \cup F \cup S) = n(E) + n(F) + n(S) - n(E \cap F) - n(E \cap S) - n(F \cap S) + n(E \cap F \cap S) $$
where $n(E)$ indicates the number of elements of $E$. So \begin{align*} &n(E \cup F \cup S) = 100 - 22 = 78\\ &n(E) = 40\\ &n(F) = 30\\ &n(S) = 25 \\ &n(E \cap F) = 8\\ &n(E \cap S) = 6\\ &n(F \cap S) = 5\\ \end{align*}
You can use this same method for similar questions.