<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 Let $a_1, a_2, \ldots, a_n$ be a permutation of $1, 2, \ldots, n$. The number of elements that are to the left of $a_i$ and smaller than $a_i$ is called the inversion number of $a_i$ ($i=1,2,\ldots,n$). For example, in the permutation $6, 4, 5, 3, 2, 1$, the inversion number of $5$ is $1$, and the inversion number of $3$ is $0$. Find the number of different permutations of the numbers $1, 2, 3, 4, 5, 6, 7, 8$ such that the inversion number of $8$ is $2$, the inversion number of $7$ is $3$, and the inversion number of $5$ is $3$. (Express the result as a number.) Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
144
Let $a_1, a_2, \ldots, a_n$ be a permutation of $1, 2, \ldots, n$. The number of elements that are to the left of $a_i$ and smaller than $a_i$ is called the inversion number of $a_i$ ($i=1,2,\ldots,n$). For example, in the permutation $6, 4, 5, 3, 2, 1$, the inversion number of $5$ is $1$, and the inversion number of $3$ is $0$. Find the number of different permutations of the numbers $1, 2, 3, 4, 5, 6, 7, 8$ such that the inversion number of $8$ is $2$, the inversion number of $7$ is $3$, and the inversion number of $5$ is $3$. (Express the result as a number.)
test