4UNIT Question :( (1 Viewer)

AbsoluteValue · Jul 2, 2012

Let m and n be two positive integers with:
$m> n$
a) Show that:
$\sum_{r=1}^{n+1} \binom{2n+1}{n+r}=2^{2n}$

b) By considering the coefficient of $x^{k}$ in $(1+x)^{m} (1+\frac{1}{x})^{n}$ show that:
$\sum_{r=0}^{m-k} \binom{m}{k+r}\binom{n}{r}=\binom{m+n}{n+k}$
I know how to do part (a) but have no idea what to do for part (b). Can anyone just explain to me what to do and I will do the rest, no need to solve it here as I know LaTex is very time consuming.

rolpsy · Jul 2, 2012

hmm..

try using:

$\left ( 1 + x \right )^m\left ( 1 + \frac{1}{x} \right )^n &= x^{-n}(1+x)^{m+n}$

and comparing coefficients

4UNIT Question :( (1 Viewer)

AbsoluteValue

Member

rolpsy

Member

Users Who Are Viewing This Thread (Users: 0, Guests: 1)