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学生在学习《排列与组合》一章时,由千对某些习题不能正确理解,从而对于解答则无法验证。因此,做出错误的解答而不知其错。此种情况有时见于某些书刊。如翻译出版的《高考数学习题集》第98页5.039题:“有30人分成三组。每组10人,共有多少种不同的分组方法?”书后答案为C_(30)~(10)·C_(20)~(10)·C_(10)~(10)(种)。很多学生做此题时也得到这个答案,此答案是否正确呢?由于此题得数较大,不易直接验证。因此,我们试用类比的方法进行研究。 类比题一:有四个人分成两组,每组两人。共有多少种不同的分组方法? 解:设四人为A,B,C,D,真分成两组。每组两人的分法有:{A、B},{C、D);{A、C},{B、D};{A、D},{B、C}。共三种。 请注意,C_4~2·C_2~3=(种) 类比题二:有六个人分成三组,每组两人、共有多少种不同的分组方法?
When students study the chapter “Arrangement and Composition”, thousands of certain exercises cannot be correctly understood, and thus the answers cannot be verified. Therefore, the wrong answer was made and it was not known. This situation is sometimes found in certain publications. Such as the translation and publication of the “College Entrance Examination Questions” on page 98 5.039 title: “There are 30 people divided into three groups. Each group of 10 people, a total of how many different grouping methods?” After the answer is C_ (30) ~ (10) · C_(20)~(10) C_(10)~(10)(species). Many students also get this answer when they do this question. Is this answer correct? Because this question has a larger number, it is not easy to directly verify. Therefore, we try analogy methods for research. Analogical question 1: There are four people divided into two groups, two in each group. How many different grouping methods are there? Solution: Let four people be A, B, C, D, divided into two groups. The division method for each group is: {A, B}, {C, D); {A, C}, {B, D}; {A, D}, {B, C}. There are three kinds. Please note that C_4~2·C_2~3=(species) Analogy Question 2: There are six people divided into three groups, each with two different grouping methods.