Question

Let n be a positive whole number. i) Prove that
nx2nCn = (n+1)x2nCn-1
ii) Use that fact that 2nCn - 2nCn-1 is a whole number to prove that 1/n+1x2nCn is a whole number

Can you advise on method please.
Thanks.

Answer 1

just write it up, it's not difficult to prove it ... although I believe you rather want combinatorial argument why?

Answer 2

for i) think about the ways forming a committee of n members from total of 2n members and selecting a caption from n members.
of think about selecting n-1 committee members first and a captain from remaining members.

Answer 3

ii)
\[ \binom{2n}{n}-\binom{2n}{n-1} = \frac{1}{n+1} \left ( (n+1)\binom{2n}{n}-(n+1)\binom{2n}{n-1}\right ) \\ = \frac{1}{n+1} \left ( (n+1)\binom{2n}{n}-(n)\binom{2n}{n}\right ) = \frac{1}{n+1} \binom{2n}{n}\]

Answer 4

I believe it is called Catlan number.
http://en.wikipedia.org/wiki/Catalan_number