66. In a Bashkir village, every resident speaks either Bashkir, or Russian, or both languages. 912 residents of the village speak Bashkir, 653 - Russian, and 435 people speak both languages. How many residents are there in this village?
$\triangle$ Let's apply Euler circles. Let $A$ denote the set of villagers who speak Bashkir, and $B$ the set of villagers who speak Russian (Fig. 26).

![](https://cdn.mathpix.com/cropped/2024_05_21_0aa0e6ac168e80891953g-032.jpg?height=308&width=468&top_left_y=1345&top_left_x=1251)

Fig. 26

![](https://cdn.mathpix.com/cropped/2024_05_21_0aa0e6ac168e80891953g-032.jpg?height=569&width=1330&top_left_y=1800&top_left_x=363)

![](https://cdn.mathpix.com/cropped/2024_05_21_0aa0e6ac168e80891953g-033.jpg?height=297&width=460&top_left_y=274&top_left_x=341)

C

10

Let's denote the number of elements in any finite set $A$ by $n(A)$. Then, according to the problem,

$$
n(A)=912, \quad n(B)=653, \quad n(A \cap B)=435
$$

We need to find the number of elements in the union of sets $A$ and $B$.

First, let's add the numbers $n(A)$ and $n(B)$. However, the elements that belong to the intersection of sets $A$ and $B$ are counted twice. Therefore, we need to subtract $n(A \cap B)$ from this sum. We get:

$$
n(A \cup B)=n(A)+n(B)-n(A \cap B).
$$

Substitute the values of $n(A)$, $n(B)$, and $n(A \cap B)$ into formula (1):

$$
n(A \cup B)=912+653-435=1130
$$

Answer: 1130.

Clearly, formula (1) is valid not only under the conditions of problem 66, but also for any finite sets $A$ and $B$.