Yep
I got 10 desirable outcomes out of a total of 32 possible options which is 10/32 = 5/16
Method:
The first digit must be 7 then there are 5 more digits with 2 posibilities each so there are 2^5 = 32 possible numbers.
If a number is divisable by 77 then it must be divisable by 7 and 11.
Since all the digits are either 0 or 7 all the numbers will be divisable by 7 so I need to worry about if it is divisable by 11.
Now I looked up the internet fo find tricks for deciding if a number is divisable by 11.
This is what I found.
https://www.math.hmc.edu/funfacts/ffiles/10013.5.shtml
So if I talk about odd and even digit places then I have
7(even), odd,even, odd, even, odd
I have to have the same number of 7s in even numbers as I do in odd ones so that they will cancel each other out.
so I can have
6 sevens one way to get this 777777
4 sevens the first even seven is set, 2 choices for second even one, only one odd one gest left out so that is 3 choices,
so I have 1*2*3 = 6 ways to have 4 sevens.
2 sevens The even one is set and there are 3 odd ones to chose from so that is 1*3 = 3
So altogether that makes 1+6+3 =10 numbers that are divisable by 77
So that is a probablility of 10/32 = 5/16