Monday, 13 February 2017

combinatorics - Using Riemann sum in a Combinatorial Proof Question

The combinatorial proof is to prove that
nk=1k(nk)=n2n1




The solution says to count the leader and committee in two ways.
The first way you choose the leader in n diff ways. Then choose the rest of the committee in 2n1 ways to get the second side of the equation.



The second way you choose a committee with k people in (nk) ways, and then there are k ways to choose its leader.
So this is suppose to give you the left side of the equation
nk=1k(nk)=n2n1



What I'm not understanding is why in the second part, you use a Riemann's sum to add all the ways. Because if you're just choosing k people, I'm not sure why the k value has to increase.



Thanks.

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