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。

  1. 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。

  2. 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。

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完整解法

6[0]

1. Identify the number of distinct units to be arranged.

By grouping the two first prize awardees as 11 unit and the two second prize awardees as another 11 unit, the remaining 44 students act as individual units.
Total units to arrange =1+1+4=6= 1 + 1 + 4 = 6 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 66 units =6!= 6!
Number of ways to arrange the first prize awardees within their group =2!= 2!
Number of ways to arrange the second prize awardees within their group =2!= 2!

Total possible arrangements =6!×2!×2!= 6! \times 2! \times 2!
=720×2×2= 720 \times 2 \times 2
=2880= 2880

答案

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 11 way to do this since the 1010 seats are identical.
This leaves 99 remaining seats: 44 on the left, 44 on the right, and 11 directly opposite the guest of honour.

2. Determine the position of the empty seat and arrange the students.

Since exactly 44 students must be seated on each side (left and right) of the guest of honour, they must occupy all 44 seats on the left and all 44 seats on the right. The seat directly opposite must therefore be the single empty seat.
The 88 students can be arranged in these 88 specific seats in 8!8! ways.
8!=403208! = 40320

答案

40320

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