$\left( 21!-21 \right):\left( 1.1!+2.2!+3.3!+...+19.19! \right)$
$=\left( 21!-21 \right):\left[ \left( 2-1 \right).1!+\left( 3-1 \right).2!+\left( 4-1 \right).3!+...+\left( 20-1 \right).19! \right]$
$=\left( 21!-21 \right):\left( 2.1!-1!+3.2!-2!+4.3!-3!+...+20.19!-19! \right)$
$=\left( 21!-21 \right):\left( 2!-1!+3!-2!+4!-3!+...+20!-19! \right)$
$=\left( 21!-21 \right):\left[ 2!+3!+4!+...+20!-1!-2!-3!-...-19! \right]$
$=\left( 21!-21 \right):\left( 20! \right)$ $=\frac{\left( 21!-21 \right)}{20!}$