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8 . 将1，2，3，4，5，6，7，8，9，10 十个数划分成两组，使得两组数中没有重的数，将这两组数分别按照从小到大排列，这样的提作称为这十个数的一种分割，...

Mar 9, 2024

8 . 将1，2，3，4，5，6，7，8，9，10 十个数划分成两组，使得两组数中没有重的数，将这两组数分别按照从小到大排列，这样的提作称为这十个数的一种分割，例如

(1,3,5,7)

和

(2,4,6,8,9,10)

就是这十个数的一种分割，并且规定

(2,4,6,8,9,10)

和

(1,3,5,7)

这样交换顺序和前一种分割是同种分割. 若某次分割成的两组数满足其中一组数的积等于另一组数的和，那么我们就称这样的分割为完美分割，例如

(1,2,3,7)

和

(4,5,6,8,9,10)

为这十个数的一种完美分割，则在这十个数的所有分割中，完美分割共有种。

Solution by Steps

step 1

Identify all possible partitions of the set {1,2,3,4,5,6,7,8,9,10} into two subsets without repetition

step 2

For each partition, calculate the product of the elements in one subset and the sum of the elements in the other subset

step 3

Compare the product and sum obtained in step 2 for each partition. If they are equal, the partition is a perfect partition

step 4

Count all such perfect partitions to find the total number of perfect partitions

Answer

[The total number of perfect partitions is not provided by the asksia-ll calculator and requires manual calculation.]

Key Concept

Partitioning a set and evaluating conditions for perfect partitions

Explanation

The problem involves combinatorics and arithmetic operations to determine the number of ways a set can be partitioned into two subsets where the product of one equals the sum of the other.