Why do we use combination, not permutation, in binomial distribution?

Question

I am aware there is a question similar to mine here and here, however I'm asking because I have a more specific question regarding this issue. Some explanations I've gotten as to why we use combination is that the order doesn't matter. HTHT and HHTT is the same number of heads. But then wouldn't this mean that the "number" of ways we get two heads is just one? Obviously not because ${4\choose2}$ is not 1.

Furthermore, can someone explain what $\frac{n!}{(n-k)!}$ would mean in the context of the "4 trials, 2 heads" example? In permutations since order matters, doesn't this mean we count HHTT, HTHT, HTTH, and so on? But then when I add this up this comes to a total of 6, which is actually ${4\choose2}$!

Answer 1

Your confusion is on the meaning of "the order doesn't matter".

When we say that there are $4\choose 2$ ways to get two tails in 4 coin flips, it's because there are $HHTT, HTHT, HTTH,$ etc. There is a first coin flip, a second coin flip etc. and in this respect, the order matters as $HT$ is not considered the same as $TH$.

However, to count the amount of ways to get $2$ tails in $4$ flips, we count the number of ways to choose $2$ placeholders from a total of $4$ possible options. To get $TTHH$, we choose the first two placeholders. And here is where the order doesn't matter: if you choose the first placeholder, and then you choose the second, or if you choose the second and then the first, in the end you get $TTHH$. In other words you don't even have to choose them one by one, you can choose your two placeholders at once.

Answer 2

If we are taking no reference to order, then there is only one way to have two heads. This is because the total number of flips must add up to $4$. Likewise, the number of ways for no heads, one head, etc. is also one for the same reason.

In the example however, you seem to have a slight mistake in your mindset. We aren't exactly choosing the number of heads with order not mattering. This is a categorization problem. It's the four distinct flips that we are categorizing into one of two categories: Heads or Tails. What doesn't matter is the order that we pick which flip gets to fall under the category of Heads. Arnaud's answer explains this in a little more detail.

An example where this sort of "Permutation vs Combination" decision comes in is problems where we have a race with, say, $12$ contestants. If I were to want to know the number of ways the top $3$ could finish with respect to first, second, and third place, then this is a Permutation problem. However, if I only care about the number of ways to make a group of the best $3$ racers, then this is a Combination problem.

Answer 3

We are measuring the probability for: some arrangement of $r$ successes among $n$ trials, each with an identical and independent rate of success of $p$.

Now the probability for a particular arrangement of such is: $p^r~(1-p)^{n-r}$.

However, since the order of the successes does not matter, all such arrangements producing that outcome must be accounted among the measure.

Now the count for distinct ways to select places to put $r$ successes among $n$ trials, to form such an arrangement, is: $\binom nr$.

Therefore the probability we seek is :$$\dbinom nr~p^r~(1-p)^{n-r}$$