We know that the sum of nine numbers is nine, so the average of these nine numbers is 1. Since all 9 numbers are positive integers, the smallest number can only be 1 and the largest number can only be 9.
We can arrange and combine the numbers one by one. First, determine the first number. Since the smallest number is 1, we can put 1 in the first place. Then, in the remaining eight positions, we can choose any number to fill in. So, there are eight possibilities.
We will determine the second number. Since the first number has been determined as 1, the sum of the remaining numbers is 8. Similarly, we can choose any number to fill the second position. There are seven other possibilities.
And so on, can be determined one by one, until all nine positions are filled.
To sum up, by permutation and combination, we can get a nine-digit password. The characteristic of this number is that it consists of nine numbers, and the sum is equal to 9. According to the calculation, there are 8X7x6x5x3x2x1x1= 40320 possible passwords.
The benefits of learning mathematics
1. Develop logical thinking ability: Mathematics is a highly logical subject. By studying mathematics, we can cultivate and develop our logical thinking ability. Mathematical problems need reasoning, analysis and solution, which cultivates students' logical thinking ability and enables them to better understand and solve various problems.
2. Cultivate problem-solving ability: Mathematics learning involves the process of solving various problems. By solving mathematical problems, students can cultivate their problem-solving ability. Mathematical problems require students to think, analyze and reason, and find methods and steps to solve the problems. This way of thinking can be applied to other disciplines and daily life.