How many different ways can you get two heads and two tails in four tosses of a fair coin?
Let's list them out:
which is 6 ways.
Another way to do this (and is actually how I caught that I'd missed 2 in the list above) is to work out the problem with a permutation formula (we care about the order of the coin flips, so that HHTT is different from TTHH. If they weren't different, like cards in a poker hand, then we'd be talking about combinations)
The general formula for a permutation is
Here we're saying that we have a population of 4 (taking each coin flip as a member of that population), picking 4 (we're doing a coin flip for all 4 of them). However, we need to adjust the formula a bit because we have only 2 results and each result appears twice. And so we divide by
So we get: