Probability question

Questions
1

I would appreciate help with this question: What is the probability that two nine digit numbers, each using all of the digits 1 through 9, will have exactly two of the digits in the same place in the number?
Thank you.
-Ron

2

Is there a Mathematica angle?

This sounds like homework.

3

Not homework, I am having a debate with my adult son and neither of us knows how to answer the question.
Thank you.
-Ron

4

We could simulate an experiment with 10^5 random pairs:

In[2]:= N[Sum[a = RandomSample[Range[9]]; b = RandomSample[Range[9]]; 
          Boole[Count[a - b, 0] == 2], {10^5}]/10^5]         

Out[2]= 0.18364

That’s not too far from the exact probability

In[3]:= (1/2) Sum[(-1)^j/j!, {j, 0, 7}]

Out[3]= 103/560

In[4]:= N[%]  

Out[4]= 0.183929

See the following link for a proof: The Matching Problem.

5

Oh my. More complex than I had thought.
Thank you very much.
-Ron