Ihave a final on this stuff tomorrow.
how many different seven-digit phone numbers exist that start with the number 5 and end with an odd number?

What's your hurry ?  The night is young.

The first digit can only be 5 ... 1 possibility.
The 2nd digit can be any one of  10.  For each of those . . .
The 3rd digit can be any one of  10.  For each of those . . .
The 4th digit can be any one of  10.  For each of those . . .
The 5th digit can be any one of  10.  For each of those . . .
The 6th digit can be any one of  10.  For each of those . . .
The 7th digit can be any one of  5.  (1, 3, 5, 7, or 9).

Total possibilities: (1 x 10 x 10 x 10 x 10 x 10 x 5) =  5 x 10⁵ = 500,000

I have no idea how many of them exist, but that's the quantity that could exist.