positive integers less than 10000

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

210 = (2)(3)(5)(7)

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

j_shreyans wrote:Hi Brent ,

Thanks for the explanation one thing more i need to clear.

(5)(7) = 35

so 1,6,35

this above case can also be formed?

Please advise and correct me if i am wrong.

Thanks,

Shreyans

Hi Shreyans,

The question says that the product of the digits is 210
35 isn't a digit, so we need not consider that case.

Cheers,
Brent

j_shreyans wrote: ↑

Fri May 15, 2015 8:47 am

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

Solution:

Note that the prime factorization of 210 = 2 x 105 = 2 x 3 x 5 x 7.

Since no two-digit numbers multiply to 210, the integer must be either 3 digits or 4 digits (if it’s less than 10,000).

If it’s a 3-digit integer, then the 3 digits must be {5, 6, 7} (notice that 5 x 6 x 7 = 210). Since there are 3! = 6 ways to permute the 3 digits, there are six 3-digit integers such that the product of their digits is 210.

If it’s a 4-digit integer, then the 4 digits must be {2, 3, 5, 7} (notice that 2 x 3 x 5 x 7 = 210) or {1, 5, 6, 7} (notice that 1 x 5 x 6 x 7 = 210). Since there are 4! = 24 ways to permute the 4 digits, there are 24 four-digit integers in each group such that the product of their digits is 210. In total, there are 24 + 24 = 48 four-digit integers such that the product of their digits is 210.

Therefore, there are 6 + 48 = 54 such integers.

Answer: D