Question 920291
{{{r*(nCr)}}}{{{""=""}}}{{{n*("(n-1)"[""]*C(r-1))}}}
<pre>
By definition {{{ACB = A!/(B!(A-B)!)}}}

{{{r*(n!/(r!(n-r)!))}}}{{{""=""}}}{{{n*((n-1)!/((r-1)!(n-1-(r-1))!))}}}

{{{r*(n!/(r!(n-r)!))}}}{{{""=""}}}{{{n*((n-1)!/((r-1)!(n-1-r+1)!))}}}

{{{r*n!/(r!(n-r)!)}}}{{{""=""}}}{{{(n*(n-1)!)/((r-1)!(n-r)!)}}}

Write the r! in the denominator on the left as r(r-1)!
Write the n*(n-1)! in the numerator on the right as n!

{{{r*n!/(r*(r-1)!(n-r)!)}}}{{{""=""}}}{{{n!/((r-1)!(n-r)!)}}}

Cancel the r's on the left:

{{{cross(r)*n!/(cross(r)*(r-1)!(n-r)!)}}}{{{""=""}}}{{{n!/((r-1)!(n-r)!)}}}

{{{n!/((r-1)!(n-r)!)}}}{{{""=""}}}{{{n!/((r-1)!(n-r)!)}}}

Both sides are now the same.

Edwin</pre>