Undergrad: Okay, just to make sure I have this straight: That's because we're considering results like HHHTHH or TTTTTT to be equally or more extreme, and there are 14 total possible cases like that, and we flipped the coin 6 times which gives us $~$2^6 \= 64$~$ possible results\. 14/64 = 22%, which is not less than 5%, so this is not statistically significant at the $~$p<0.05$~$ level\.

"That's because we're considering results like HHHTHH or TTTTTT to be equally or more extreme, and there are 14 total possible cases like that."

It is not evident for me, because I am not familiar with statistics. I think where is a need to provide calculation, that there is two examples in form TTTTTT and HHHHHH and 6*2 like HTHHHH or HTTTTT.