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How many ways can 3 digits be combined to multiply out to 24?

1 x 3 x 8 - 1 x 4 x 6 - 2 x 2 x 6 - 2 x 3 x 4

Each of these can be combined in a 6 different ways:

1 x 3 x 8 - 1 x 8 x 3 - 3 x 1 x 8 - 3 x 8 x 1 - 8 x 3 x 1 - 8 x 1 x 3

1 x 4 x 6 - 1 x 6 x 4 - 4 x 1 x 6 - 4 x 6 x 1 - 6 x 1 x 4 - 6 x 4 x 1

2 x 2 x 6 - 2 x 6 x 2 - 6 x 2 x 2 (the other three are repeats)

2 x 3 x 4 - 2 x 4 x 3 - 3 x 2 x 4 - 3 x 4 x 2 - 4 x 2 x 3 - 4 x 3 x 2

So we have 21 different numbers whose digits multiply out to 24:

138, 183, 318, 381, 813 and 831

146, 164, 416, 461, 614 and 641

226, 262 and 622

234, 243, 324, 342, 423 and 432

Q: How many 3-digit positive integers have digits have a product of 24?

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