Question 1107228
Part I
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{{{ C(n,r) = n!/((n-r)!(r!)) }}}  (by definition)
{{{ (n/(n-r))C(n-1,r) = (n/(n-r))((n-1)!/((n-1-r)!(r!))) }}}
 = {{{ (n*(n-1)!)/((n-r)(n-r-1)!*r!)) }}} 
We can bring the n into the factorial in the numerator to go from (n-1)! to n!, 
and similarly we can bring in the {{{cross(n-1)}}} (EDIT: n-r) in the denominator to go from (n-1-r)! to (n-r)! :
 = {{{  n!/((n-r)!*r!) }}}   DONE.
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Part II
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LHS:  nC(n-1,r) = {{{ n*(n-1)!/((n-1-r)!*r!) = n!/((n-1-r)!*r!) }}}
RHS:  (r+1)C(n,r+1) = {{{ (r+1)*n!/((n-r-1)!(r+1)!) }}}
 = {{{ (r+1)*n!/((n-r-1)!(r+1)(r!)) }}}
Canceling (r+1) from numerator and denominator:
= {{{ n!/((n-r-1)!*r!) }}} = LHS,   DONE