Question 947191
Question 940228
<pre>
nC(r-1) + nC(r) ?=? (n+1)Cr
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(1)   nCr = {{{n!/(r!(n-r)!)}}}

(2)   nC(r-1) = {{{n!/((r-1)!(n^""-(r-1))!))}}} = {{{n!/((r-1)!(n-r+1)!)}}}

(3)   (n+1)Cr = {{{(n+1)!/(r!((n+1)^""-r)!))}}} = {{{(n+1)!/(r!(n+1-r)!)}}}

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We want to prove that expression(2) + expression(1) = expression(3)

nC(r-1) + nCr =

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

Substitute (n-r+1)(n-r)! for (n-r+1)! and r(r-1)! for r!

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

LCD = r(r-1)!(n-r+1)(n-r)!

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

replace r(r-1)! by r! and replace (n-r+1)(n-r) by (n-r+1)!

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

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

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

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

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

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

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

(n+1)Cr

Edwin</pre>