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?
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.
+2666 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?
