Question 1089122
The expansion of a binomial {{{(a+b)^n}}} has {{{n+1}}} terms.
They are of the form {{{Ca^pb^q}}} with a combinatorial coefficient {{{C}}} ,
and positive integers {{{p}}} and {{{q}}} such that {{{p+q=n}}} .
There's is a certain symmetry to the combinatorial coefficients:
The first and last ones (the number {{{1}}} and {{{n+1}}} coefficients are 1;
the number {{{2}}} and {{{n}}} coefficients are {{{n}}} ;
the number {{{3}}} and {{{n-1}}} coefficients are {{{n(n-1)/2}}} ,
and so on, so that the number {{{m}}} and {{{s}}} coefficients will be the same if {{{m+s=n+2}}} .
So, in this case,
{{{(2r+1)+(3r+2)=21+2}}} ,
{{{5r+3=23}}} , 
{{{5r=20}}} ,
and {{{highlight(r=4)}}} .
 
Verification:
{{{2r+1=2*4+1=8+1=9}}}
{{{3r+2=3*4+2=12+2=14}}}
The 9th coefficient is
{{{(matrix(2,1,21,8))=21!/((21-8)!8!)=21!/(13!8!)=203490}}} .
The 14th coefficient is
{{{(matrix(2,1,21,13))=21!/((21-13)!13!)=21!/(8!13!)=203490}}} .