500 followers problem

To have a product equal to 12, the three digits must be (1,2,6), (1,3,4) or (2,2,3) so 126, 162, 216, 261, 612, 621, 134, 143, 314, 341, 413, 431, 223, 232, 322 are the three digit numbers having a product that is equal to 12. But the problem gave 126 and asked for OTHER three digit numbers so 126 should be excluded. There are fourteen of them

The possible values of digits are (1,2,6), (1,3,4), and (2,2,3). (1,2,6) can be orientated 3! ways, (1,3,4) can also be orientated 3! ways, and (2,2,3) can be orientated (3!/2!) ways. 6+6+3 = 15. Subtract 1 for the example given to get 14 possible three digit numbers.

The numbers are 126,162,216,261,612,621 134,143,314,341,413,431 223,232,322.So excluding 126,we have 14 such numbers

To create a product of 3 number equal to 12 we must have $\ (1,2,6), (1,3,4) or (2,2,3)$ . For the first 2 triples we have 3! + 3! = 12 choices. However, in the 3rd triples 2 appears twice so there are 3 choices (just moving the 3). However we must exclude (1,2,6) , so there are 12 + 3 - 1 = 14 choices

12 has factors 2x2x3x1. Therefore, making 3 digits number out of these factors we get, 126 with its 3!==6 permutations,431 with its 3!==6 permutations and 223 with its 3!/2!==3 permutations. So , in total we got 15 such numbers and after discarding the given one i.e., 126, we have left with 14 such numbers .

But 126 shold be excluded according to the question and the answer should be 14