Question 1175056
<pre>

1) nCr = n!/((n-r)!r!)   <<< this is what we need to show for the RHS

  RHS: C(n+1,r) - C(n,r-1)   where C(a,b) = "aCb"
   
 = {{{ (n+1)!/((n+1-r)!r!) -  n!/((n-(r-1))!(r-1)!)  }}}

Factor out n+1 from the first term, and multiply the 2nd term by r/r:
 = {{{ ((n+1)n!)/((n+1-r)!r!) -  (n!*r)/((n-(r-1))!(r)!)  }}}

Notice (n+1-r)! ( = (n-(r-1))! ) is the same as  (n+1-r)(n-r)!

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


Factor out  n!/((n-r)!r!) and re-write:

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

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

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

= {{{ nCr }}}  &#9632;



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(2) nPr = n!/(n-r)! = P(n,r)  <<< P(x,y) notation used below

    (n-r+1)*P(n,r-1)  

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

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

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

=    {{{ nPr }}}  &#9632;