Ten empty boxes are arranged in a row, and there are eight balls. The balls are to be placed into the boxes so that no boxes are left unused, the total number of balls is , and each box contains at most two balls. (The balls are indistinguishable.) Find the number of such distributions that satisfy the following conditions. [4 points]\begin{center}\fbox{\begin{minipage}{0.9\textwidth}(A) The number of boxes containing exactly one ball is either or .(B) Any box containing two balls has empty neighboring boxes.\end{minipage}}\end{center}\
[THOUGHT] We need to count assignments of 0,1,2 to 10 positions (boxes) such that:
- Total sum of values = 8 (since each ball contributes 1, each box with 2 balls contributes 2).
- The number of positions with value 1 is either 4 or 6.
- For any position with value 2, both its neighbors (if they exist) must be 0. We can brute‑force all 3^10 = 59049 possibilities using itertools.product, filter by the constraints, and sum the counts for the two allowed values of k (4 and 6). This will give the exact number of distributions. [/THOUGHT] [PYTHON] from itertools import product
def is_valid(arr): # arr is a tuple of length 10 with values 0,1,2 # check sum if sum(arr) != 8: return False # count of 1's count1 = arr.count(1) if count1 not in (4, 6): return False # check condition for 2's n = len(arr) for i, v in enumerate(arr): if v == 2: # left neighbor if i > 0 and arr[i-1] != 0: return False # right neighbor if i < n-1 and arr[i+1] != 0: return False return True
total = 0 for assignment in product([0,1,2], repeat=10): if is_valid(assignment): total += 1
print(total) [/PYTHON]
[THOUGHT] The Python code exhaustively checked all 3^10 = 59049 possible assignments of 0, 1, or 2 balls to the 10 boxes. It filtered these assignments based on the two conditions:
- The total number of balls is exactly 8.
- The number of boxes with exactly one ball is either 4 or 6.
- For any box containing two balls, both adjacent boxes (if they exist) must be empty. The result from the code execution was 262, which is the exact count of valid distributions satisfying all constraints. [/THOUGHT]
\boxed{262}