Hey, firstly yes your C(CH3)3 means the carbon atom is connected to 3 methyl groups.
Then as for your question regarding the naming, first you have to select the longest carbon chain possible. When u select the longest chain and number the C atoms,u should number them in a way that the sum of the numbers assigned to C atoms bearing functional groups is minimum.
I'll now explain with the example.
In the above compound, the longest carbon chain can be the one starting from the Church attached to three methyl groups and towards the left corner of the compound.Then before naming u should number all C atoms.If u start numbering from the left most corner, u will get 2+4+4+6+7+7=30 as the sum of the numbers assigned to C atoms bearing functional groups. But if u start numbering from the C atom that bears three methyl groups, u will get, 2+2+3+5+5+7=24 as the sum of numbers assigned to C atoms bearing functional groups.As u can see, 30>24. So u should number the C chain starting from the C atom attached to three methyl groups.
Now for the naming.
When naming u should start from the functional group of which its first letter comes first in the alphabet.So then it will be exactly the answer that u were given with.I have uploaded a photo for u to get a clear idea of the above explanation.