Resolução:
Veja que:
$\require{cancel}$$\phantom{X}5\centerdot 5!\;=\;(6 - 1)\centerdot 5!\;=$ $\;6\centerdot 5!\,-\,5!\;=\;(5\,+\,1)!\,-\,5!\phantom{X}$
então:
$\phantom{X}(5\,+\,1)!\,-\,5!\,+\,(6\,+\,1)!\,-\,6!\;...\;(19\,+\,1)!\,-\,19!\;=\;$$\;\cancel{6!}\,-\,5!\,+\,\cancel{7!}\,-\,\cancel{6!}\;...\;+\,20!\,-\,\cancel{19!}\;=\,20!\,-\,5!\phantom{X}$