Query about question "A man flips a fair coin with sides heads and tails five times..." [99]

GRE GRE Quant Problem Solving
1

https://www.gregmat.com/problems/quant?problem=a-man-flips-a-fair-coin-with-sides-heads-and-tails-five-times

For this question, how do you know how to calculate the possibilities across all five flips of the coin instead of just counting out two “heads” flips and finding the probability that there are exactly two “tails” flips in the remaining three flips? I understand that the two given “heads” flips could happen at any points in the five flips, but the order isn’t important, so why not treat those as extraneous events and just look at the three flips that are in question?

Thanks for your help!

2

The order actually IS important while considering the total number of probable cases.

HHHTT and HTHTH is not the same set of flips (both have 3 heads and 2 tails). If you treat them extraneous, then your total number of cases will reduce, and your probability will shoot up.