鸽巢原理 容斥原理.ppt

2006级
2008. 11. 9
鸽巢原理与容斥原理
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Contents
鸽巢原理：简单形式
鸽巢原理：加强形式
容斥原理
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Pigeonhole Principle
If 7 pigeons are to live in 6 boxes (holes), then there is at least one box containing two or more pigeons.
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Dirichlet (狄里克雷)
“鸽巢原理”，最先是由19世纪的德国数学家狄里克雷应用于解决问题，后人们为了纪念他从这么平凡的事情中发现的规律，就把这个规律用他的名字命名，叫“狄里克雷原理”，又把它叫做“抽屉原理”。
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Pigeonhole Principle
If we put n+1 balls into n boxes, then at least one box must contain two or more balls.
将n+1个球放入n个盒子内, 最小有一个盒子藏有2个或以上的球。
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Proof
We want to prove that ‘at least one box must contain two or more balls’.
反证法
Assume that the statement above is wrong then all the n boxes contains 1 or 0 ball. Therefore the total number of balls is less than or equal to n. This is a contradiction (矛盾). The contradiction is due to our assumption that the ‘statement is wrong’.
We conclude that the statement must be true.
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Examples
For any 368 people, at least two of them must have the same birthday.
There are at least two people in the world having same number of hair.
At least two of you in this class (assuming the class size >12) born in the same month.
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Exercise
There are 10 married couples. How many of the 20 people must be selected in order to guarantee that one has selected a married couple ?
Answer: 11
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Pigeonhole Principle : Strong Form (鸽巢原理－加强形式)
If we put (k*n+1) balls in n boxes, then
at least one box must contain k+1 or
more balls.
将 (k*n+1) 个球放入n个盒子內, 最小有一个盒子藏有k+1个或以上的球。
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Proof
We want to prove that at least one box must contain k+1 or more balls.
Assume that the statement above is wrong then all the n boxes contains k or less balls. Therefore the total number of balls is less than or equal to n*k. This is a contradiction (矛盾). The contradiction is due to the assumption that ‘the statement is wrong’.
We conclude that the statement must be true.
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