How can the instances in the private subnet connect to the Internet?
Use the internet gateway with a private IP
Allow outbound traffic in the security group for port 80 to allow internet updates
The private subnet can never connect to the internet
Use NAT with an elastic IP
Explanations:
An internet gateway is only used with public subnets. Instances in a private subnet cannot directly connect to the internet using a private IP address because they do not have a public IP.
Allowing outbound traffic in the security group for port 80 will not enable instances in a private subnet to connect to the internet. Private subnets lack direct internet access regardless of security group settings.
While instances in a private subnet cannot connect to the internet directly, they can connect indirectly via a NAT gateway or NAT instance. Thus, this statement is misleading.
Using a NAT (Network Address Translation) gateway with an elastic IP allows instances in a private subnet to connect to the internet for tasks like patch updates. The NAT gateway handles the outbound traffic and allows the responses to return to the instances.
I conclude that the answer is:
Use NAT with an elastic IP