组合数学 (Fall 2019)/Problem Set 1: Difference between revisions

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== Problem 1 ==
== Problem 1 ==
箱子里有<math>m</math>个相同红球,同时有其他<math>n</math>种不同的颜色的球各一个。若从箱中一口气取出r个球,有多少种可能?(不考虑取的先后,只考虑最后的结果。)
Suppose that there are <math>n</math> red balls, and another m balls are distinct and not red. Balls with the same color are indistinguishable.
#<math>r\leq m,r\leq n</math>
Determine the number of ways to select <math>r</math> balls from these <math>n+m</math> balls, in each of the following cases:
#<math>n\leq r\leq m</math>
#<math>r\leq m,r\leq n</math>;
#<math>m\leq r\leq n</math>
#<math>n\leq r\leq m</math>;
#<math>m\leq r\leq n</math>.
 
== Problem 2 ==
李雷和韩梅梅竞选学生会主席,韩梅梅获得选票 <math>p</math> 张,李雷获得选票 <math>q</math> 张,<math>p>q</math>。我们将总共的 <math>p+q</math> 张选票一张一张的点数,有多少种选票的排序方式使得在整个点票过程中,韩梅梅的票数一直高于李雷的票数?等价地,假设选票均匀分布的随机排列,以多大概率在整个点票过程中,韩梅梅的票数一直高于李雷的票数。
 
== Problem 3 ==
Find the generating function for the sequence <math>(a_r)</math> in each of the following cases: <math>a_r</math> is the number of ways of distributing <math>r</math> identical objects into:
#4 distinct boxes;
#4 distinct boxes so that no box is empty;
#4 identical boxes;
#4 identical boxes so that no box is empty.
 
== Problem 4 ==

Revision as of 13:56, 16 September 2019

To be constructed

  • 每道题目的解答都要有完整的解题过程。中英文不限。

Problem 1

Suppose that there are [math]\displaystyle{ n }[/math] red balls, and another m balls are distinct and not red. Balls with the same color are indistinguishable. Determine the number of ways to select [math]\displaystyle{ r }[/math] balls from these [math]\displaystyle{ n+m }[/math] balls, in each of the following cases:

  1. [math]\displaystyle{ r\leq m,r\leq n }[/math];
  2. [math]\displaystyle{ n\leq r\leq m }[/math];
  3. [math]\displaystyle{ m\leq r\leq n }[/math].

Problem 2

李雷和韩梅梅竞选学生会主席,韩梅梅获得选票 [math]\displaystyle{ p }[/math] 张,李雷获得选票 [math]\displaystyle{ q }[/math] 张,[math]\displaystyle{ p\gt q }[/math]。我们将总共的 [math]\displaystyle{ p+q }[/math] 张选票一张一张的点数,有多少种选票的排序方式使得在整个点票过程中,韩梅梅的票数一直高于李雷的票数?等价地,假设选票均匀分布的随机排列,以多大概率在整个点票过程中,韩梅梅的票数一直高于李雷的票数。

Problem 3

Find the generating function for the sequence [math]\displaystyle{ (a_r) }[/math] in each of the following cases: [math]\displaystyle{ a_r }[/math] is the number of ways of distributing [math]\displaystyle{ r }[/math] identical objects into:

  1. 4 distinct boxes;
  2. 4 distinct boxes so that no box is empty;
  3. 4 identical boxes;
  4. 4 identical boxes so that no box is empty.

Problem 4