23) E

In order for the number to consist of 3 digits, one of the numbers must be a two digit number and the second number must be a single digit number. 

All single digit numbers = 1, 2, 3, 4, 5, 6, 7, 8, 9
All double digit numbers = 10, 11, 12, 13, 14, 15

Remember that ALL DIGITS MUST BE DIFFERENT. So 1 cannot be matched with 10, 11, 12, 13, 14 or 15 as they all have 1. Similarly, 11 cannot be match with anything as it has two 1s.

2 can only be matched with 10, 13, 14, 15
3 can only be matched with 10, 12, 14, 15
4 can only be matched with 10, 12, 13, 15
5 can only be matched with 10, 12, 13, 14
6 can only be matched with 10, 12, 13, 14, 15
7 can only be matched with 10, 12, 13, 14, 15
8 can only be matched with 10, 12, 13, 14, 15
9 can only be matched with 10, 12, 13, 14, 15

Total number of possibilities of having three digits all different = (4 x 4) + (5 x 4) = 16 + 20 = 36

Now we must look at total number of possible pairings.

1 can be matched with 14 other numbers.
2 can be matched with 13 other numbers. (13 because 2 and 1 would have already had 1 pair).
Similarly, number of possibilities:
14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 105

So probability = (36 / 105) = (12 / 35)

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