Principle of Inclusion and exclusion

We suppose there is a list of property and we assume there is a certain subset of elements that have the property and the rest of the elements do not. We want to get the number of elements that have none of the properties:

Let $N_{0} = n$
Count the number of objects having each property which is $(\binom{n}{1})$ , and let $N_{1}$ be the result
In general look at the $(\binom{k}{i})$ possible subsets of $i$ properties. For each of these we count the number of objects haaving all $i$ properties in the subset. THen add up all these numbers call it $N_{i}$

Theorem: Suppose given k properties and given an n-set X let $N_{0}, . . ., N_{k}$ be as defined above. Then the number M of elements with none of the properties is given by the formula:
$M = \sum_{i = 0}^{k} (- 1)^{i} N_{i}$