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

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

by BTGmoderatorDC » Tue Nov 23, 2021 6:54 pm

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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

OA D

Source: Magoosh

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

by swerve » Wed Nov 24, 2021 2:40 pm
BTGmoderatorDC wrote:
Tue Nov 23, 2021 6:54 pm
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

OA D

Source: Magoosh
$$210 = 2\times5\times3\times7 = 5\times6\times7\times1 = 5\times6\times7$$

Those are the only sets of digits we can use to for the numbers (any other combination of factors will have two digit factors).

Numbers using $$2,5,3,7 = 4!$$
Numbers using $$5,6,7,1 = 4!$$
Numbers using $$5,6,7$$ (3-digit numbers) $$= 3!$$

Answer $$= 24+24+6 = 54$$

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

by [email protected] » Thu Nov 25, 2021 6:43 am
BTGmoderatorDC wrote:
Tue Nov 23, 2021 6:54 pm
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

OA D

Source: Magoosh
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