In how many ways can 12 white and 12 black checkers be placed on the black squares of a chessboard?

#
First, we need to choose 12 fields for the white checkers, and then from the remaining 20 fields, choose 12 fields for the black checkers.

## Answer

$$
C_{32}^{12} \cdot C_{20}^{12}=\frac{32!}{12!\cdot 12!\cdot 8!} \text { ways. }
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_16504755117b31808ad7g-08.jpg?height=169&width=1054&top_left_y=230&top_left_x=16)

How many five-digit numbers exist that read the same from left to right and from right to left (for example, such as 54345, 17071)?

## Hint

It is sufficient to specify the first three digits of such a number - the rest will be uniquely determined.

## Answer

$9 \cdot 10^{2}=900$ numbers.