The teacher asked: Does everyone understand the exchange of treasures between the seven brothers of Hulu?
Two people in unison, I understand!
Teacher Yi said: Learning mathematics means learning ideas and methods; We must not be satisfied with answering one or two specific exercises correctly; Be sure to thoroughly understand the principle of solving problems.
If you don't know the basic principles deeply enough and the proposer changes the specific figures, you may get the wrong answer.
Fangfang replied: Teacher, we really understand. Don't believe me, you test us with a question.
Teacher Yi: OK, I'll ask a similar question now. You try, can you answer correctly?
Two primary school students are working hard at writing and handing in their papers at the same time in a few minutes.
Among them, Fang Fang's answer is this.
The eight immortals exchange treasures, and the exchange schemes can be divided into the following three categories.
In the first category, the Eight Immortals are divided into eight groups. According to the cyclic arrangement mode, the specific number of exchange schemes is:
In the second category, the Eight Immortals are divided into two groups, a group of three people and a group of five people.
For this category, there are three steps:
(1) The number of schemes grouped is;
(2) The number of exchange schemes for the trio is
(3) The number of exchange schemes for a five-person group is
According to the principle of multiplication, we can find that the total number of such schemes is;
In the third category, the Eight Immortals can be divided into two groups, each with four people.
Similar to the second category, the number of projects in the third category is
According to addition principle, the total number of eight immortals exchange schemes is equal to the sum of three schemes, namely:
Teacher Yi: Sisi, is your conclusion the same as Fangfang's?
Sisi: Same.
Teacher Yi said with a smile: It seems that everyone has basically mastered the multiplication principle, the multiplication principle. Fang Fang's thinking and general direction of solving problems are correct.
But now let's go deeper. We know that one way is to choose four people from eight people. This number is a bit large. Can you write some concrete examples? Sisi, you write.
Sisi's students quickly wrote the following example:
Very good!
Teacher Yi continued: Since there are two groups, we might as well call the group written by Fang Fang Group A and the other group B ... Sisi, write down the corresponding members of Group B.
According to the teacher's request, Sisi quickly chose another group of members and got the following table.
Teacher Yi: Look at the table above carefully. What did you find?
After watching them for a while, Sisi suddenly said, I found that the penultimate and the first are actually the same grouping scheme.
Fang Fang went on to say: More than one pair. The same is true of the first and second types. Each grouping scheme appears twice in this table.
Teacher Yi: That is to say, in our previous calculations, there were repeated statistics.
Fangfang: I remember. This has happened before. On the question of handshake, it actually represents the same handshake. So, it should be divided by 2.
Teacher Yi: Exactly. In the third exchange, the correct number of packets should be.
So, the total number of exchange schemes of the Eight Immortals is:
Teacher Yi: Learning mathematics means learning principles, ideas and methods. Mathematical problem solving can't stay on the surface, and it must be analyzed and summarized in depth to make progress.
Specific to the application of addition principle, multiplication principle, permutation and combination, there are some subtle differences, which must be noted.
In the calculation of grouping, sometimes it is necessary to divide by 2, and sometimes it is not necessary. Can you give two simple examples to illustrate?
According to the teacher's request, two students made up a question respectively.
Dear readers, do you see the difference between the above two questions?