Probability, factorials, the mystery of it all

Question

Continuing on with the GRE practice questions I have, I'm confused about how to solve the following types of probability questions.

1. Martha invited 4 friends to go with her to the movies. There are 120 different ways in which they can sit together in a row of 5 seats, one person per seat. In how many of those ways is Martha sitting in the middle seat?
   
   There are 120 ways they can sit together in a row of 5 seats because 5! = 120. I'm confused as to how you would mathematically describe a situation where Martha is sitting only in the "middle" seat.

2. How many 3-digit positive integers are odd and do not contain the digit 5 ?
   
   If I am correct (probably not), there are 999–99=900 3-digit positive integers (this means there are 450 odd 3-digit positive integers, I think). Of these, 100/900 will be a 5 (i.e. 5xx), 10/100 10s will be a 5 (i.e. x5x), and 1/10 1s digits will be a 5 (i.e. xx5). At this point I'm lost in numbers and it has taken me much longer than the minute I will be given to solve it. Am I missing some sort of heuristic to solving these?

Thanks for you help.

Answer 1

1. Fix the middle seat for Martha. Now you have to position the other 4 people in 4 seats: $4!=24$.
2. Build the number by the digits: you have $8$ options for the hundreds (all but 0 and 5), $9$ options for the tens (all but 5) and $4$ options for the units (only one of $1,3,7,9$). A total of $8\cdot 9\cdot 4$ numbers.