How many positive integers less than 10,000 are such that the product of their digits is 210?
a) 24
b) 30
c) 48
d) 54
e) 72
a) 24
b) 30
c) 48
d) 54
e) 72
210 = (2)(3)(5)(7)How many positive integers less than 10,000 are such that the product of their digits is 210?
(A) 24
(B) 30
(C) 48
(D) 54
(E) 72
We need to consider 3 cases:
Case 1: 4-digit numbers using 2, 3, 5, 7
There are 4 digits, so this can be accomplished in 4! (24) ways
Aside: Notice that (2)(3) = 6
Case 2: 4-digit numbers using 1, 6, 5, 7
There are 4 digits, so this can be accomplished in 4! (24) ways
Case 3: 3-digit numbers using 6, 5, 7
There are 3 digits, so this can be accomplished in 3! (6) ways
Add up all 3 cases to get 24 + 24 + 6 = 54
So, the answer is D
Cheers,
Brent
