组合数学 (Fall 2015)/Problem Set 1: Difference between revisions
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imported>Etone (Created page with "<font color="red" size=5>每道题目的解答都要有完整的解题过程。中英文不限。</font> == Problem 1 == #有<math>k</math>种不同的明信片，每种明信…") |
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== Problem 2 == | == Problem 2 == | ||
Find the number of ways to select <math>2n</math> balls from <math>n</math> identical blue balls, <math>n</math> identical red balls and <math>n</math> identical green balls. | |||
== Problem 3 == | |||
* Give a combinatorial proof for the following identity: | * Give a combinatorial proof for the following identity: | ||
:<math> | :<math> | ||
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*Give a closed form for <math>\sum_{k=1}^n k^2{n\choose k}</math> and prove it. | *Give a closed form for <math>\sum_{k=1}^n k^2{n\choose k}</math> and prove it. | ||
==Problem | ==Problem 4== | ||
A <math>2\times n</math> rectangle is to be paved with <math>1\times 2</math> identical blocks and <math>2\times 2</math> identical blocks. Let <math>f(n)</math> denote the number of ways that can be done. Find a recurrence relation for <math>f(n)</math>, solve the recurrence relation. | A <math>2\times n</math> rectangle is to be paved with <math>1\times 2</math> identical blocks and <math>2\times 2</math> identical blocks. Let <math>f(n)</math> denote the number of ways that can be done. Find a recurrence relation for <math>f(n)</math>, solve the recurrence relation. | ||
Revision as of 02:11, 21 September 2015
每道题目的解答都要有完整的解题过程。中英文不限。
Problem 1
- 有[math]\displaystyle{ k }[/math]种不同的明信片，每种明信片有无限多张，寄给[math]\displaystyle{ n }[/math]个人，每人一张，有多少种方法？
- 有[math]\displaystyle{ k }[/math]种不同的明信片，每种明信片有无限多张，寄给[math]\displaystyle{ n }[/math]个人，每人一张，每个人必须收到不同种类的明信片，有多少种方法？
- 有[math]\displaystyle{ k }[/math]种不同的明信片，每种明信片有无限多张，寄给[math]\displaystyle{ n }[/math]个人，每人收到[math]\displaystyle{ r }[/math]张不同的明信片（但不同的人可以收到相同的明信片），有多少种方法？
- 只有一种明信片，共有[math]\displaystyle{ m }[/math]张，寄给[math]\displaystyle{ n }[/math]个人，全部寄完，每个人可以收多张明信片或者不收明信片，有多少种方法？
- 有[math]\displaystyle{ k }[/math]种不同的明信片，其中第[math]\displaystyle{ i }[/math]种明信片有[math]\displaystyle{ m_i }[/math]张，寄给[math]\displaystyle{ n }[/math]个人，全部寄完，每个人可以收多张明信片或者不收明信片，有多少种方法？
Problem 2
Find the number of ways to select [math]\displaystyle{ 2n }[/math] balls from [math]\displaystyle{ n }[/math] identical blue balls, [math]\displaystyle{ n }[/math] identical red balls and [math]\displaystyle{ n }[/math] identical green balls.
Problem 3
- Give a combinatorial proof for the following identity:
- [math]\displaystyle{ \sum_{k=1}^n k{n\choose k}=n\cdot 2^{n-1}. }[/math]
- Give a closed form for [math]\displaystyle{ \sum_{k=1}^n k^2{n\choose k} }[/math] and prove it.
Problem 4
A [math]\displaystyle{ 2\times n }[/math] rectangle is to be paved with [math]\displaystyle{ 1\times 2 }[/math] identical blocks and [math]\displaystyle{ 2\times 2 }[/math] identical blocks. Let [math]\displaystyle{ f(n) }[/math] denote the number of ways that can be done. Find a recurrence relation for [math]\displaystyle{ f(n) }[/math], solve the recurrence relation.