# Thread: Inclusion-exclusion math problem

1. ## Inclusion-exclusion math problem

I am stumped on this question.

"How many elements are in the union of four sets if each of the sets has 100 elements, each pair of the sets shares 50 Elements, each three of the sets share 25 elements, and there are 5 elements in all four sets?"

Thanks in advance for the help.

2. [Hello, rspierz!

How many elements are in the union of four sets
if each of the sets has 100 elements,
each pair of the sets shares 50 elements,
each three of the sets share 25 elements,
and there are 5 elements in all four sets?
You could draw a Venn diagram
. . (if you know how to draw one with four sets).

You should be familiar with these two Union Formulas:

$[1]\;\;n(A \cup B) \;=\;n(A) + n(B) - n(A \cap B)$

$[2]\;\;n(A\;\cup\;B\;\cup\;C) \;=\;n(A) \;+\; n(B) \;+\; n(C) \;-\; n(A\;\cap\;B) \;-\; n(B\;\cap\;C) \;-\; n(A\;\cap\;C)$ $\;+\; n(A\,\cap\,B\,\cap\,C)$

But you may not be aware of this one:

$n(A \cup B \cup C \cup D) \;=\;n(A) + n(B) + n(C) + n(D)$

. . . . . . . . . . . . . . $- n(A\cap B) - n(A \cap C) - n(A\cap D) - n(B\cap C) - n(B\cap D) - n(C \cap D)$

. . . . . . . . . . . . . . . . $+ n(A \cap B \cap C) + n(A \cap B \cap D) + n(A \cap C \cap D) + n(B \cap C\cap D)$

. . . . . . . . . . . . . . . . . . $- n(A \cap B \cap C \cap D)$