1. Consider the permutations $a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}$ formed by $\{1,2,3,4,5,6\}$, which can be transformed into 123456 with no more than 100 operations. Find the number of permutations that satisfy this condition.
(12th Korean Mathematical Olympiad)
Solution: For any permutation $a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}$, take $a_{1}$. If $a_{1}$ is not in position $a_{1}$, take the number $a_{1}^{\prime}$ in position $a_{1}$; if $a_{1}^{\prime}$ is not in position $a_{1}^{\prime}$, take the number $a_{1}^{\prime \prime}$ in position $a_{1}^{\prime}$, and continue until the number $a_{1}$ in position $a_{1}$ is taken. Then $\left(a_{1}, a_{1}^{\prime}, a_{1}^{\prime}, \cdots, a_{1}^{\prime}\right)$ is called a chain. The length of the chain is the number of elements in the chain. Clearly, the length of the chain can be $1,2,3,4,5,6$, and a chain of length $i(i=1,2,3,4,5,6)$ requires $i-1$ swaps to achieve the goal, until all chains are divided into chains of length 1.
Now we solve this problem using the exclusion method.
Since the problem requires no more than 4 swaps, the longest chain in a permutation that meets the condition can be of length $5,4,3,2,1$, and any permutation with the longest chain of length $5,4,3,2,1$ meets the condition.
Now we examine the number of permutations with the longest chain of length 6, so the required number is $6!-5!=720-120=600$.
(Zhao Jiwei, Class 002, Zhengzhou No.1 High School, Zhengzhou, Henan Province, China)