Yes, you need to write test cases for each and every 
permission. I think there should be multiple test cases for 
each user to further see if by logging in with a particular 
User ID, User is able to do the required functions on the 
application.

For Example: If user id has read only/ view only 
permissions than user should not be able to do any 
modifications.

If user has admin permissions than he can do 
X,Y,Z....modifications...Thus, for admin user there will be 
multiple test cases...

No,you need not to write test cases for each and every 
permission.According to me Single Test case refer to 170
permissions. But depend upon the type of USER.
Example: For Admin User need to write 6 Test Case that under
cones in multiple user also that is created by admin .
         For Normal User only 4 Test Case.

Test Cases can be clubbed as Anjali is saying but then you 
will have a very long script.Imagine running 1 test case 
for 5 days and if I user fail than entire TC is failed nad 
you would need to reexcute the entire TC. I think this will 
be too much of an overhead.

First of all you have to have the clear requirements that 
those 170 permissions will be used by how many users, like:

user1, user2, user3, user4 etc...
Then based on the users you have to categorize the 
permissions, that login as user1 who has permission1, 
permission2, etc and verify so and so, etc... by this way 
you can reduce the number of test cases by writing each 
test cases for each permissions i.e. (170 permissions).