Since #n! = 1 times 2 times ... times (n-1) times n#, we have

#{17!}/{5!times 12!} = {1 times 2 times ...times 17}/{(1 times 2times ...times 5)(1 times 2 times ... times 12)}#

# = {color(red)(1 times 2 times ...12) times 13 ... times 17}/{(1 times 2times ...times 5)(color(red)(1 times 2 times ... times 12))}#

# ={13 times 14 times color(red)15 times color(blue)16 times 17}/{1 times color(blue)2 times color(red)3 times color(blue)4 times color(red)5} = 13times 14 times 2 times17 = 6188#