2. Find the number of all seven-digit natural numbers for which the digits in the decimal representation are in strictly increasing order up to the middle, and then in strictly decreasing order. For example, the number 1358620 would fit. 

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# Solution.

If the middle digit of a seven-digit number is 4 (it cannot be less), the number of desired numbers can be obtained by erasing one digit to the right of the " " "middle digit of the final number" in the number 12343210. There are 4 such numbers.

If the middle digit of a seven-digit number is 5, the number of desired numbers can be obtained by erasing one digit to the left of the "5" - the middle digit of the final number - and two digits to the right in the number 1234543210. There are $C_{4}^{1} C_{5}^{2}=4 \cdot 10$ such variants.

Similarly, if the middle digit of a seven-digit number is 6, the number of "suitable" ones is $C_{5}^{2} C_{6}^{3}=10 \cdot 20$.

For a middle digit of 7, there will be $C_{6}^{3} C_{7}^{4}=20 \cdot 35$ such numerical variants.

For 8 and 9, we will get, respectively, $C_{7}^{4} C_{8}^{5}=35 \cdot 56$ and $C_{8}^{5} C_{9}^{6}=56 \cdot 84$.

In total $4+4 \cdot 10+10 \cdot 20+20 \cdot 35+35 \cdot 56+56 \cdot 84=$

$$
=4 \cdot 11+20 \cdot(10+35)+56 \cdot(84+35)=7608 . \text { Answer: } 7608 .
$$

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