By doing port scanning is it possible to hack into a network

By doing port scanning is it possible to hack into a network wherein
*person has a /29 cidr block of ip's
*person is using a firewall cum router say a pix
*person has 5 static ip's (public ip's) which he has configured in the pix
*he has forced the private ip's from the NAT of the firewall/router in the tcp/ip properties of 5 individual pc's by shutting the dhcp off on the firewall cum router
*he has mapped the public ip's with the private ip's on the firewall in such a way that only authorised people can access the pc's from remote locations

now my question is if a hacker wants can he infiltrate this security
by doing port scanning
there are logics which I am unable to understand like
*how can we shut the dhcp off and still do a ip mapping on the pix
*there are 65000 ports available on the pc and a hacker for doing a port scan has to specify a range.......what is the probability that he can breach the security
*and even if he does a hacker can only do port scanning on public ips and not private ip's (is this true?)
*can a hacker do a port scanning and get to know the private ip's as well

pls reply soon.........I would be anxiously waiting for the answer
thank you all

how can we shut the dhcp off and still do a ip mapping on the pix

DHCP is used to dynamically distribute network settings on the hosts of a network, while IP mapping is commonly used to transparently redirect (translate) traffic between two or more IP addresses, in the case that some of them are not routed to the other's network. In essense, dhcp only complicates things by creating an extra factor to consider -that of dynamicly assigned ip addresses. Without having used a PIX, I believe that static addresses would only make easier to configure what you want.

there are 65000 ports available on the pc and a hacker for doing a port scan has to specify a range.......what is the probability that he can breach the security

There are 65356 ports available and a hacker could easily scan all of them -though it could take days if he wanted to take caution against triggering an alert to Intrusion Detection/Prevention Systems that monitor for activity usual to scanning (in case one does exist). Otherwise -depending on the network speed and the tcp/ip stack of the involved hosts, he could probe all the ports in a few minutes. By default, only a set of "standard" ports is usually scanned, where common services listen to (you can find the list here ). The probability that the security can be breached is not easy to estimate since there are a lot of factors that need to be considered, the ability to establish connections to some port is just the most obvious factor and an -even properly- configured firewall is just one measure of protection.

and even if he does a hacker can only do port scanning on public ips and not private ip's (is this true?)

Port scanning is done by sending IP packets to a host of the network. In Internet's case, hosts of the network are only hosts with public IP addresses, since the private ones are not routeable.

can a hacker do a port scanning and get to know the private ip's as well

No. The private addresses can only be contacted through NAPT / IP mapping to some public one(s), and that is a transparent procedure: the remote hosts only know that they contact the public address.

Unfortunatelly network security as a whole is not that straight-forward as your (or any in fact) questions. If you want to secure a network with confidence you should first understand how TCP/IP works and then read some good books that will give you an idea of the possible threats -though with a good knowledge of tcp/ip and a good imagination, you'll have an idea of what to expect.