Question 1077089
<pre>{{{matrix(1,3,nC12,""="",nC8)}}}

If you know that {{{matrix(1,3,nCr,""="",nC(n-r))}}}, then
it's easy.

r = 12,  n-r = 8
        n-12 = 8
           n = 20

Otherwise:

{{{matrix(1,3,n/(12!(n-12)!),""="",n!/(8!(n-8)!))}}}

Cross-multiply:

{{{matrix(1,3,n!(8!(n-8)!),""="",n!(12!(n-12)!))}}}

Divide both sides by n!

{{{matrix(1,3,8!(n-8)!,""="",12!(n-12)!)}}}

Write 12! as 12&#8729;11&#8729;10&#8729;9&#8729;8!

{{{matrix(1,3,8!(n-8)!,""="",12*11*10*9*8!(n-12)!)}}}

Divide both sides by 8!

{{{matrix(1,3,(n-8)!,""="",12*11*10*9(n-12)!)}}}

Write (n-8)! as (n-8)&#8729;(n-9)&#8729;(n-10)&#8729;(n-11)&#8729;(n-12)!

{{{matrix(1,3,(n-8)*(n-9)*(n-10)*(n-11)*(n-12)!,""="",12*11*10*9(n-12)!)}}}

Divide both sides by (n-12)!

{{{matrix(1,3,(n-8)*(n-9)*(n-10)*(n-11),""="",12*11*10*9)}}}

n-8=12, n-9=11, n-10=10, n-11=9
  n=20,   n=20,    n=20,    n=20

Answers: {{{n = 20}}},  {{{nC17 = 20C17 = 20C3 = (20*19*18)(3*2*1)=1140}}}

Edwin</pre>