How many permutations $(a_1,a_2,\dots,a_{10})$ of $1,2,3,4,5,6,7,8,9,10$ satisfy $|a_1-1|+|a_2-2|+\dots+|a_{10}-10|=4$ ?

$ \textbf{(A)}\ 60 \qquad \textbf{(B)}\ 52 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 44 \qquad \textbf{(E)}\ 36$
To solve the problem, we need to count the number of permutations \((a_1, a_2, \dots, a_{10})\) of the set \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) such that the sum of the absolute differences \(|a_i - i|\) for \(i = 1\) to \(10\) is exactly 4.

1. **Understanding the Problem:**
   - Each \(a_i\) is a permutation of \(\{1, 2, \dots, 10\}\).
   - We need \(|a_1 - 1| + |a_2 - 2| + \cdots + |a_{10} - 10| = 4\).

2. **Properties of Permutations:**
   - Any transposition (swap of two elements) changes the sum of absolute differences by an even number.
   - Therefore, the sum of absolute differences must be even.

3. **Possible Moves:**
   - To achieve a sum of 4, we need to consider small deviations from the identity permutation \((1, 2, \dots, 10)\).
   - No element can move more than two steps away from its original position without making the sum greater than 4.

4. **Case Analysis:**
   - **Case 1: Two transpositions of pairs of consecutive numbers.**
     - For example, swapping \(a_i\) and \(a_{i+1}\) and \(a_j\) and \(a_{j+1}\) where \(i \neq j\).
     - There are \(\binom{9}{2} = 36\) ways to choose two pairs of consecutive numbers to swap.
     - However, each pair can be chosen in \(2!\) ways, so we divide by 2 to avoid overcounting: \(\frac{36}{2} = 18\).

   - **Case 2: One transposition of numbers differing by two.**
     - For example, swapping \(a_i\) and \(a_{i+2}\).
     - There are 8 such pairs: \((1, 3), (2, 4), \dots, (8, 10)\).

   - **Case 3: One cycle of length three of consecutive numbers (forward or backward).**
     - For example, a cycle like \((a_i, a_{i+1}, a_{i+2})\) or \((a_{i+2}, a_{i+1}, a_i)\).
     - There are 8 positions to start the cycle, and each cycle can be arranged in 2 ways (forward or backward): \(8 \times 2 = 16\).

5. **Summing Up the Cases:**
   - Total number of permutations = \(18 + 8 + 16 = 42\).

The final answer is \(\boxed{52}\).