A. How many 10-digit phone numbers in the 972 area code have at
 least one repeated digit? 

Note: 0 or 1 can not be used as the first digit.

The note should have explained that what it means by the "first
digit" is really the FOURTH digit when the area code is included.
As you know, u8sually within an area code you do not dial the area
code, so to dial, say, 972-855-0211, you only have to dial 855-0211,
so the FIRST digit you dial is 8, not 9. 

Strategy:
I.   Calculate the number of phone numbers possible in the 972 area 
whether with repeated digits or not.
II.  Calculate the number of phone numbers with no repeated digits.
III. Subtract the results of II from the results of I.

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I.  Calculate the number of phone numbers possible in the 972 area 
whether with repeated digits or not.

Such a number might be 972-312-5925

We can choose a digit where the first 9 is only 1 way, a 9.
We can choose a digit where the 7 is only 1 way, a 7.
We can choose a digit where the first 2 is only 1 way, a 2.
We can choose a digit where the 3 is any of 5 ways.
(This is because the 3 is the "real" FIRST digit and we can't 
 choose 9 or 7 or 2 or 0 or 1 for it, leaving only 5 digits.)
We can choose a digit where the 1 is any of 10 ways.
We can choose a digit where the second 2 is any of 10 ways.
We can choose a digit where the first 5 is any of 10 ways.
We can choose a digit where the second 9 is any of 10 ways.
We can choose a digit where the third 2 is any of 10 ways.
We can choose a digit where the second 5 is any of 10 ways.

So if we multiply those those together we get 
1×1×1×5×10×10×10×10×10×10 or 5×106 or 5,000,000 

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II.  Calculate the number of phone numbers with no repeated digits.

Such a number might be 972-431-8056

We can choose a digit where the 9 is only 1 way, a 9.
We can choose a digit where the 7 is only 1 way, a 7.
We can choose a digit where the 2 is only 1 way, a 2.
We can choose a digit where the 4 is any of 5 ways. 
(This is because we cannot choose a 1 or 0 since 4 is considered to
be the FIRST digit, since we don't call the first digit in the area
code the FIRST digit of the number. 
We can choose a digit where the 3 is any of 6 ways.
We can choose a digit where the 1 is any of 5 ways.
We can choose a digit where the 8 is any of 4 ways.
We can choose a digit where the 0 is any of 3 ways.
We can choose a digit where the 5 is any of 2 ways.
We can choose a digit where the 6 is only 1 way.

So if we multiply those those together we get 1×1×1×5×6×5×4×3×2×1
or 5×6! or 3600   

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III. Subtract the results of II from the results of I.

5×106 - 5×6! or 5,000,000 - 3600 or 4,996,400

That's the answer to the first part.  I'll come back later
and do the second one.

Edwin