Counting routes from home to office if you only return home if you realize you have forgotten something before you reach the office
The following is how you can go to the office from home:
Home Four Roads Schools Three Roads University Five Roads Parks Two Roads Offices
You are forgetful minded. You may have forgotten something at home. You remember what you forgot at home, either at your school or university or in the park, and you go back to pick it up. Then the journey continues from the beginning again. You forget one thing at most during the day and when you reach the office you do not go back to take back what you have left. So how many different routes are possible for you?
I have tried this way:
Case 1: I remember what I forgot at school:
In this case, I can go to school in 4 ways and come back in 4 ways and go back in 4 ways. Then I can start the journey from there in ways.
Case 2: I remember what I forgot at university:
In this case, I can go to university in ways and come back in ways and go back in ways. Then I can start the journey from there in ways.
Case 3: I remember what I forgot at park:
In this case, I can go to park in ways and come back in ways and go back in ways. Then I can start the journey from there in 2 ways.
Case 4: I go to office without remembering:
In this case, I can go to the office in ways.
So, total ways would be ways.