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# Counting problem

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The product of the digits of 3214 is 24. How many distinct four-digit positive integers are such that the product of their digits equals 24?

Jun 4, 2022

### 2+0 Answers

#1
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1234 , 1243 , 1324 , 1342 , 1423 , 1432 , 2134 , 2143 , 2314 , 2341 , 2413 , 2431 , 3124 , 3142 , 3214 , 3241 , 3412 , 3421 , 4123 , 4132 , 4213 , 4231 , 4312 , 4321 , Total =  24 such 4-digit distinct integers.

Jun 4, 2022
#2
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The only ways to get 24 is if we have $$1, 2, 3, 4$$$$1,1,3,8$$, or $$1,1,4,6$$

For the first case, because all the digits are distinct, there are $$4! = 24$$ ways to factor.

For the second case, we have to account for overcounting by dividing by $$2!$$(the amount of ways to order 2 1s), giving us: $${4 ! \over 2!} = 12$$

LIkewise, the third case has 12, by the same logic.

Now, can you take it from here?

Jun 4, 2022