For a permutation $\pi$ of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where $\pi (i)$ denotes the $i$th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers from 1 to 10. Determine the number of permutations $\pi$ for which $S(\pi) = M$.

[i]Ray Li[/i]
To solve the problem, we need to analyze the expression for \( S(\pi) \) and determine the maximum possible value of \( S(\pi) \) over all permutations of the integers from 1 to 10. Then, we need to count the number of permutations that achieve this maximum value.

1. **Expression Analysis**:
   \[
   S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1)) \cdot (4 + \pi(i) + \pi(i+1))
   \]
   We can rewrite the expression inside the sum:
   \[
   (\pi(i) - \pi(i+1)) \cdot (4 + \pi(i) + \pi(i+1))
   \]
   This can be expanded as:
   \[
   (\pi(i) - \pi(i+1)) \cdot 4 + (\pi(i) - \pi(i+1)) \cdot (\pi(i) + \pi(i+1))
   \]
   Simplifying further:
   \[
   4(\pi(i) - \pi(i+1)) + \pi(i)^2 - \pi(i+1)^2
   \]

2. **Sum Simplification**:
   The sum of the differences of squares can be simplified using the telescoping property:
   \[
   \sum_{i=1}^{9} (\pi(i)^2 - \pi(i+1)^2) = \pi(1)^2 - \pi(10)^2
   \]
   The sum of the linear terms:
   \[
   \sum_{i=1}^{9} 4(\pi(i) - \pi(i+1)) = 4(\pi(1) - \pi(10))
   \]

3. **Combining Terms**:
   Combining the simplified sums, we get:
   \[
   S(\pi) = 4(\pi(1) - \pi(10)) + (\pi(1)^2 - \pi(10)^2)
   \]
   Factoring the quadratic term:
   \[
   S(\pi) = 4(\pi(1) - \pi(10)) + (\pi(1) - \pi(10))(\pi(1) + \pi(10))
   \]
   \[
   S(\pi) = (\pi(1) - \pi(10))(4 + \pi(1) + \pi(10))
   \]

4. **Maximizing \( S(\pi) \)**:
   To maximize \( S(\pi) \), we need to maximize the product \((\pi(1) - \pi(10))(4 + \pi(1) + \pi(10))\). The maximum difference \(\pi(1) - \pi(10)\) occurs when \(\pi(1) = 10\) and \(\pi(10) = 1\):
   \[
   S(\pi) = (10 - 1)(4 + 10 + 1) = 9 \cdot 15 = 135
   \]

5. **Counting Permutations**:
   To achieve the maximum value \( S(\pi) = 135 \), \(\pi(1)\) must be 10 and \(\pi(10)\) must be 1. The remaining 8 elements can be permuted in any order:
   \[
   \text{Number of permutations} = 8!
   \]

The final answer is \(\boxed{40320}\)