[THOUGHT] The problem involves distributing 8 indistinguishable balls into 10 distinguishable boxes in a row, each box getting 0, 1, or 2 balls (sum to 8). The phrase "no boxes are left unused" leads to impossibility if no empties allowed (since 8<10), so empty boxes are permitted; likely poor wording. There are two separate conditions: (A) number of boxes with exactly 1 ball is 4 or 6; (B) every box with 2 balls has empty adjacent boxes (both neighbors if exist, handled for edges). Use itertools.product to enumerate all 3^10 configurations, filter sum==8, and count satisfying each condition separately. [/THOUGHT] [PYTHON] import itertools

Part A

countA = 0 for config in itertools.product([0,1,2], repeat=10): if sum(config) == 8: num_ones = sum(1 for x in config if x == 1) if num_ones in (4, 6): countA += 1

Part B

countB = 0 for config in itertools.product([0,1,2], repeat=10): if sum(config) == 8: valid = True for i in range(10): if config[i] == 2: if i > 0 and config[i-1] != 0: valid = False break if i < 9 and config[i+1] != 0: valid = False break if valid: countB += 1

print("Part A:", countA) print("Part B:", countB) [/PYTHON]

[THOUGHT] The Python output provides the counts: 3990 for (A) and 462 for (B). These are the answers for the two parts. [/THOUGHT] \boxed{3990 462}

Explanation

The reasoning begins by analyzing the problem constraints: distribute 8 indistinguishable balls into 10 distinguishable boxes, with each box holding at most 2 balls and the total summing to 8. The phrase "no boxes are left unused" implies no empty boxes, but this is impossible since at least 10 balls would be needed for 10 boxes with at least one ball each. Thus, empty boxes are permitted, likely due to a wording issue, and the code proceeds by enumerating valid configurations.

To solve, generate all possible configurations using itertools.product([0,1,2], repeat=10), which produces 3^10 tuples representing box contents, then filter those summing to 8.

For part (A), count configurations where the number of boxes with exactly one ball is 4 or 6. Iterate through valid configurations (sum == 8), count boxes with 1 ball using sum(1 for x in config if x == 1), and increment the counter if the count is 4 or 6. This yields 3990.

For part (B), count configurations where every box with 2 balls has empty neighbors. For each valid configuration (sum == 8), check each box i: if config[i] == 2, verify config[i-1] == 0 (if i > 0) and config[i+1] == 0 (if i < 9). If all such boxes satisfy this, increment the counter. This yields 462.

The Python code verifies these counts exhaustively, treating boxes as distinguishable and balls as indistinguishable, aligning with the problem. The outputs—3990 for (A) and 462 for (B)—are the exact numbers of distributions satisfying each condition separately, as confirmed by the tool execution.