How many positive integers less than 10,000 are such that the product of their digits is 210?

210=1*2*3*5*7=1*6*5*7 (2*3 = 6)

Those are the only sets of numbers that can be used (any other combination of factors will have two digit answers).

Numbers using 2,5,3,7 = 4! = 4 x 3 x 2 = 24

Numbers using 5,6,7,1 = 4! = 4 x 3 x 2 = 24

Numbers using 5,6,7 = 3! = 3 x 2 = 6

24+24+6 = 54