某项竞赛中,有八名学生入围获奖名单。其中两名学生获得一等奖,另两名学生获得二等奖。 这八名入围学生将排成一排,与一位特邀嘉宾一起合影留念。
如果合影期间两名一等奖得主必须站在一起,且两名二等奖得主也必须站在一起,求有多少种不同的排法。
合影结束后,八名学生前往茶歇处,他们将在一张有 10 个相同座位的圆桌旁就座。如果特邀嘉宾两旁各坐 4 名学生,求所有可能的座位安排总数。
它不会直接告诉你答案,而是问你卡在哪一步看看 AI 是怎么陪你一步步弄懂这道题的
完整解法
6[0]
1. Identify the number of distinct units to be arranged.
By grouping the two first prize awardees as unit and the two second prize awardees as another unit, the remaining students act as individual units.
Total units to arrange units.
2. Calculate the total number of arrangements by multiplying the permutations of the units by the internal permutations of the groups.
Number of ways to arrange the units
Number of ways to arrange the first prize awardees within their group
Number of ways to arrange the second prize awardees within their group
Total possible arrangements
2880
6[1]
1. Fix the guest of honour to resolve the circular permutation and identify the available seats.
Fix the guest of honour at one seat. To account for the circular arrangement, there is way to do this since the seats are identical.
This leaves remaining seats: on the left, on the right, and directly opposite the guest of honour.
2. Determine the position of the empty seat and arrange the students.
Since exactly students must be seated on each side (left and right) of the guest of honour, they must occupy all seats on the left and all seats on the right. The seat directly opposite must therefore be the single empty seat.
The students can be arranged in these specific seats in ways.
40320
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