The symbol
$$n\choose k$$
is a shorthand for "The number of ways to choose $k$ objects from a collection of $n$ objects."
That is: you have $n$ objects in a line. You select $k$ of them, and paint them red, and you paint the other $n-k$ blue. How many ways are there to do that so that the final result looks different?
A moment of reflection will convince you that you could have chosen $n-k$ of the objects to paint blue first, and painted the rest of the objects red. The answer shouldn't depend on something arbitrary like whether you pick up the blue or the red paint first. If you can grok this fact, then you will have proved to yourself that
$${n\choose k} = {n\choose n-k}$$
How does this relate to binomial expansions? Well, say that we want to expand
$$(r + b)^n$$
Then we'll get a bunch of terms, each of which is a product of some $r$s with some $b$s. There are $n$ brackets, so there will be $n$ multipliers in each of the terms, and each of them will be either $r$ or $b$, so every term looks like
$$a_kr^kb^{n-k}$$
where $a_k$ is a coefficient, and $k=0, 1, 2, \dots n$.
What is $a_k$? Notice that in every bracket, we have to choose either an $r$ or a $b$. We can imagine going through the brackets one by one and making these choices, or you can imagine looking at all the brackets at once, and choosing which of them to pick $r$ form and which to pick $b$ from. The coefficient $a_k$ is the number of different ways of picking $k$ $r$s and $(n-k)$ $bs$ from the $n$ brackets. That is:
$$a_k = {n\choose k}$$
which, using the fact from earlier, we can instantly use to prove that the coefficients in a binomial expansion are symmetric:
$$a_k = {n\choose k} = {n\choose n-k} = a_{n-k}$$
Maybe you can see how binomial expansions relate to Pascal's triangle now?
Hint: It is linked to another cool and exciting relationship between the binomial coefficients!
In the expansion of $(a+x)^n$, the general term, that is the $(r+1)^{th}$ term is $$T_{r+1}=\binom nr a^{n-r}x^r.$$
Thus, in the expansion of $(1+x)^{23}$, the general term is $$\binom{23}r1^{23-r}x^r.$$
You want the coefficient of $x^7$, that is $r=7$, so the the coefficient is $$\binom{23}7.$$
This is the way, using general terms, you can derive any coefficient of any term without any expansion.
Best Answer
With multinomial theorem, we want the coefficient on $x^4$ terms. There are two terms that contain $x^4$. These are:
With coefficients respectively of: $$3\cdot 2\cdot \binom {12} {10,1,1}=\frac{6\cdot12!}{10!\cdot 1!\cdot 1!}=792$$ $$3^4\cdot\binom {12} {8,4,0}=\frac{81\cdot12!}{8!\cdot 4!\cdot 0!}=40095$$ Which then add to give your coefficient of $x^4$ of $40095+792=40887$