Question

Alina 把数字 $1,2,\cdots,9$ 分别写在卡片上，每张卡片一个数。她想把这些卡片分成 $3$ 组，每组 $3$ 张，使每组数字之和相同。有多少种分法？

Alina writes the numbers $1,2,\cdots,9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?

$0$

$1$

$2$

$3$

$4$

答案：C

知识点：分拆与有序分拆分类讨论

难度评级：1690

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文字解答：

所有数字的总和为 $\dfrac{9 \cdot 10}{2} = 45$ ，所以每组和必须为 $45 \div 3 = 15$ 。

含 $9$ 的组中，另外两个数和为 $6$ 。可能是 $1$ 和 $5$ ，或 $2$ 和 $4$ 。

情况 $1$ ：一组是 $1, 5, 9$ 。

考虑含 $8$ 的那组，另外两个数必须和为 $7$ 。唯一选择是 $3$ 和 $4$ 。

剩下的一组是 $2,6$ 和 $7$ ，其和为 $15$ ，所以这种情况贡献一种分法。

情况 $2$ ：一组是 $2, 4, 9$ 。

同理，含 $8$ 的那组另外两个数和为 $7$ 。唯一选择是 $1$ 和 $6$ 。

最后一组是 $3, 5$ 和 $7$ ，其和为 $15$ 这给出另一种分法。

所有情况都已检查，共有 $2$ 种分法。

所以正确答案是 C。

The sum of all the numbers is $\dfrac{9 \cdot 10}{2} = 45.$ This means that the sum of each group is $45 \div 3 = 15.$

Consider the group with $9$ in it. The other two numbers must add to $6.$ Therefore, the other cards in this group are $1$ and $5$ or $2$ and $4.$

Case $1:$ One group is $1, 5, 9$

Consider the group with $8$ in it. The other numbers must add to $7.$ The only option is $3$ and $4$ with the remaining cards.

The other group is then $2,6,$ and $7.$ This adds to $15,$ so this case contributes one possibility.

Case $2:$ One group is $2, 4, 9$

Consider the group with $8$ in it. As above, the other numbers have to add to $7.$ The only option is $1$ and $6.$

The final group is $3, 5,$ and $7,$ which adds to $15.$ This is another configuration.

We have gone through all the cases, which revealed that there are only $2$ possible groupings.

Thus, C is the correct answer.

2023 AMC 8 第 21 题

答案：C

难度评级：1690

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其他年份的第 21 题