Problem Analysis #29

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로그인 회원가입 You are solving a Korean CSAT mathematics problem. Reason carefully, but the final response must contain exactly one final answer inside \boxed{}.

For multiple-choice problems, put only one of ①, ②, ③, ④, ⑤ inside \boxed{}. For short-answer problems, put only the final integer inside \boxed{}. Do not use external tools.

Problem 30 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 8, 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] (A) The number of boxes containing exactly one ball is either 4 or 6. (B) Any box containing two balls has empty neighboring boxes. Mathematics (Calculus) 더보기

The problem states that ten boxes are arranged in a row, eight indistinguishable balls are placed such that no box is left unused (every box gets at least one ball), each box holds at most two balls, and condition (A) says the number of boxes with exactly one ball is either 4 or 6. Condition (B) says any box containing two balls has empty neighboring boxes.

First, note that “no boxes are left unused” means every box must contain at least one ball. With 10 boxes and 8 balls, this is impossible because the minimum number of balls needed is 10. Therefore, there are no distributions that satisfy the basic setup, regardless of conditions (A) and (B).

Thus, the number of such distributions is 0.

\boxed{0}

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