Help counting

The number of ways that you can arrange four different digits is  4 x 3 x 2 x 1  =  24 ways.

There are 64 ways for the product of a 4-digit number to equal 24

6 x 4 x 1 x 1 = 12 numbers

6 x 2 x 2 x 1 = 12 numbers

4 x 3 x 2 x 1 = 24 numbers

8 x 3 x 1 x 1 = 12 numbers

3 x 2 x 2 x 2 = 4 numbers

Add all of them up, and you get 64.

because you guys had different answers, I did it myself and I found idk ( I didn't count) numbers of which 24 numbers are distinct.

1138, NO
1183, NO
1146, NO
1164, NO

1226, NO
1262, NO 
1234, --------------------------- 1
1243, --------------------------- 2

1318, NO
1381, NO
1324, ---------------------------- 3
1342, ----------------------------- 4

1416, NO
1461, NO
1423, ------------------------------ 5
1432, ------------------------------ 6

1614, NO
1641, NO
1622, NO

1813, NO
1831, NO

2126, NO
2162, NO
2134, ----------------------------- 7
2143, ----------------------------- 8

2216, NO
2261, NO
2223, NO
2232, NO

2314, -------------------------------- 9
2341, -------------------------------- 10
2322, NO

2413, -------------------------------- 11
2431, -------------------------------- 12

2612, NO
2621, NO 

3118, NO
3181, NO
3124, -------------------------------- 13
3142, --------------------------------- 14

3214, -------------------------------- 15
3241, -------------------------------- 16
3222, NO

3412, -------------------------------- 17
3421, --------------------------------- 18

3811, NO

4116, NO
4161, NO
4123, ---------------------------------- 19
4132, ------------------------------------ 20

4213, ------------------------------------- 21
4231, ------------------------------------- 22

4312, ------------------------------------ 23
4321, ------------------------------------ 24

4611, NO

6114, NO
6141, NO
6122, NO

6212, NO
6221, NO

6411, NO

8113, NO
8131, NO

8311, NO

This was just like a check that geno's answer was correct... .

thus there are 24 ways.