Coin flipping

Since nobody else has published a solution yet...

This question is basically just asking you to find a way to avoid double counting:

Case 1 : First five coins are heads: $\frac{1}{32}$ chance of occurring

Case 2 : Second five coins are heads: $\frac{1}{32}$ chance of occurring... but then we double count the ones where the first 6 coins are all heads. In fact there's a neat little trick to avoid this double counting: We suppose the first flip gives tails in case 2 as this then makes it mutually exclusive to Case 1. Now this case gives $\frac{1}{64}$

Case 3,4,5,6 third set of five, etc. up to 6th set of five coins, each with a tails at the start. These four cases gives $\frac{4}{64}$

Now we add the chances together to get $\frac{7}{64}$ with 7+64=71

Now try coin flipping 2