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Homework II-A Combinatorics

 
 
ZhangEthan的头像
Homework II-A Combinatorics
ZhangEthan - 2020年01月26日 Sunday 11:27
 
I don't know how to do a combinatorial proof for questions 6.22 and 6.23, though I know the algebraic one.  Does anybody know how to do them, especially 6.22?
 
ProfessorAreteem的头像
Re: Homework II-A Combinatorics
ProfessorAreteem - 2020年01月27日 Monday 11:03
 

The idea is to give two ways of counting the same thing, one that would yield the left side of the equation and the other the right side of the equation.

For example, on 6.22 we can think of choosing two disjoint groups of size $k$ and $j$ out of $n$ objects. As we can see on the left side of the equation, one way of counting is first choosing $k$ of the $n$ objects, and then choose $j$ of the remaining $n-k$ objects.