## Task 2 - 190812

From the digits $0,1, \ldots, 9$, exactly seven are to be selected, none of which are the same.

Determine the number of (in the decimal system) seven-digit numbers that contain each of the selected digits in their (decimal) digit representation!

Assume that

a) the 0 does not occur among the selected digits,

b) the 0 does occur among the selected digits.
The mentioned seven-digit numbers must contain each of the given digits exactly once.

In case a), there are exactly 7 possibilities for filling the first position. For each of them, there remain exactly 6 possibilities for filling the second position. Therefore, there are a total of $7 \cdot 6$ possibilities for filling the first two positions.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0706.jpg?height=49&width=1596&top_left_y=2494&top_left_x=230)

In case b), there are exactly 6 possibilities for filling the first position: the further considerations proceed as in case a). Therefore, one obtains a total of $6 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=4320$ possibilities.

The sought numbers are thus a) 5040; b) 4320.