\section*{Task 3 - 300923}

a) How many three-digit natural numbers, where (like 921) the tens digit is greater than the units digit but less than the hundreds digit, are there in total?

b) How many six-digit numbers can be formed in total by selecting two different numbers from those described in a) and writing the larger of these two numbers behind the smaller one?

c) The smallest of all the six-digit numbers described in b), where the second of the mentioned three-digit numbers is exactly 1 greater than the first, is the telephone number of the sender Potsdam. What is it?
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a) Every selection of 3 different digits without considering the order leads to exactly one such three-digit number. Therefore, there are \(\binom{10}{3}=\frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1}=120\) such three-digit numbers.

b) Every selection of 2 different such three-digit numbers without considering the order yields exactly one such six-digit number. Therefore, there are exactly \(\binom{120}{2}=\frac{120 \cdot 119}{2 \cdot 1}=60 \cdot 119=7140\) such six-digit numbers.

c) To make the number as small as possible, the first digits should be as small as possible. The minimum possible value for a three-digit number from a) is 210. However, the number that is 1 greater is not of the form described in a).

For the increased unit digit to still be smaller than the tens digit, the tens digit must be at least 2, and thus the hundreds digit must be at least 3. In fact, the smallest possibility for this is 320321, so this is the desired phone number.