we have a monic polynomial $p(x)$ of degree $d$ such that
$p(k) = d-k-1,~\forall k = 0,1,\dots, d-1$
Show $p(d) = d!-1$
$q(x) = p(x) - (d-1-x)$
$q(x) = 0,~\forall x = k=0,1,\dots ,(d-1)$
$q(x) = \prod \limits_{k=0}^{d-1}~(x-(d-1-k))$
now let $d=2008$
$q(x) = \prod \limits_{k=0}^{2007}~(x-2007+k)$
$p(2008) = q(2008) +(2007-2008) = q(2008)-1$
$q(2008) = \prod \limits_{k=0}^{2007}~k+1 = 2008!$
$p(2008)=2008!-1$