1. ## Combinatorics Word Problem!

30. Last year at the Independent Learning Centre, a group of 48 students enrolled in mathematics, French, and physics. Some students were more successful than others: 32 passed French, 27 passed physics, and 33 passed mathematics; 26 passed French and mathematics, 26 passed physics and mathematics, and 21 passsed French and physics; 21 passed French, mathematics, and physics. How many students passed one or more of the subjects?

The question asks for a Venn Diagram (out of 6 marks) - which is something that I can do myself.
However, what I need help with is to come up with the principle (of inclusion or exlusion) - this is out of 2 marks.

If someone could help me come up with the principle that would be awesome! thanks

2. Hello, cnmath16!

30) Forty-eight students were enrolled in mathematics, French, and physics.

32 passed French
27 passed physics
33 passed math
26 passed French and math
26 passed physics and math
21 passsed French and physics
21 passed all three

How many students passed one or more of the subjects?
We want: .$\displaystyle n(F \cup P \cup M)$

Formula: .$\displaystyle n(F \cup P \cup M) \;=\;\begin{Bmatrix}n(F) + n(P) + n(M) \\ - n(F \cap P) - n(P \cap M) - n(F \cap M) \\ + n(F \cap P \cap M) \end{Bmatrix}$

We are given: .$\displaystyle \begin{array}{ccc}n(F) &=& 32 \\ n(P) &=& 27 \\ n(M) &=& 33\end{array}\quad \begin{array}{ccc} n(F \cap P) &=& 21 \\ n(P \cap M) &=& 26 \\ n(F \cap M) &=& 26\end{array} \quad \begin{array}{ccc}n(F \cap P \cap M) &=& 21 \end{array}$

Therefore: .$\displaystyle n(F \cup P \cup M) \;=\;32 + 27 + 33 - 21 - 26 - 26 + 21 \;=\;\boxed{40}$