Question 143
Let $a_1, a_2, \dots, a_{11}$ be $11$ distinct positive integers with their sum less than $2007$, and write the numbers $1, 2, \dots, 2007$ in order on the blackboard. Now we define a group of $22$ ordered operations:
The $i$-th operation is to take any number on the blackboard, and then add $a_i$ to it, if $1 \le i \le 11$ or minus $a_{i-11}$ from it, if $12 \le i \le 22$.
If the final result after such a group of operations is an even permutation of $1, 2, \dots, 2007$, then we call it a "good" group; if the result is an odd permutation of $1, 2, \dots, 2007$, then we call it a "second good" group.
Our question is: Which is greater? The number of "good" groups or that of "second good" groups? And by how many more?
The $i$-th operation is to take any number on the blackboard, and then add $a_i$ to it, if $1 \le i \le 11$ or minus $a_{i-11}$ from it, if $12 \le i \le 22$.
If the final result after such a group of operations is an even permutation of $1, 2, \dots, 2007$, then we call it a "good" group; if the result is an odd permutation of $1, 2, \dots, 2007$, then we call it a "second good" group.
Our question is: Which is greater? The number of "good" groups or that of "second good" groups? And by how many more?