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例题 将8个“+”号和6个“-”号排成一排,求这些符号恰好变化5次的排列种数。 错解:先将6个“-”号排成一排,并从中间的5个空档中选出2个插入两块隔板“1”,将其任意分成有顺序的3组,这时有C_5~3种分组方法。其次再将8个“+”号排成一排,这时有9个空档,我们将上述分成3组的6个“-”号中的前两组插入“+”中间7个空档中的2个,再把剩下的一组放到首末的空档中的一个,这时有C_7~2C_2~1种插法,且这些符号恰好变化5次,故总的排法种数为C_5~2C_7~2C_2~1=420种。
Example: Put eight “+” signs and six “-” signs in a row and find that these symbols change exactly by five times. Misunderstanding: First arrange six “-” signs in a row, and select two of the five intermediate slots to insert two spacers “1” and divide them into three groups in sequence. There are C_5~3 grouping methods. Then we put eight “+”s in a row. At this time there are nine gaps. We insert the first two groups of the six “-”s in the above three groups into the middle seven gaps of “+”. Of the two, and then put the remaining group in the gap between the first and the end, there are C_7~2C_2~1 kinds of interpolation, and these symbols happen to change 5 times, so the total number of rows is C_5~2C_7~2C_2~1=420 species.