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] Model each distribution as a length-10 sequence with entries 0,1,2 (balls per box). Enforce total balls = 8 and max per box = 2. Apply condition (A): number of 1’s is 4 or 6. Apply condition (B): every 2 must have neighbors (if any) equal to 0. Count sequences satisfying both, and also break down by the two cases for verification. [/THOUGHT] [PYTHON] import itertools
def valid(seq): # total balls and max already ensured by seq in {0,1,2} if sum(seq) != 8: return False ones = seq.count(1) if ones not in (4, 6): return False # condition B: any 2 has empty neighboring boxes n = len(seq) for i, v in enumerate(seq): if v == 2: if i-1 >= 0 and seq[i-1] != 0: return False if i+1 < n and seq[i+1] != 0: return False return True
count_total = 0 count_case = {(2,4):0, (1,6):0} # (twos, ones) for seq in itertools.product([0,1,2], repeat=10): if valid(seq): count_total += 1 count_case[(seq.count(2), seq.count(1))] += 1
print(count_total) print(count_case) [/PYTHON]
[THOUGHT] The Python count of valid distributions satisfying both (A) and (B) is 262, with 150 in the (4 ones, 2 twos) case and 112 in the (6 ones, 1 two) case. \boxed{262}