How does Combination formula relates in getting the coefficients of a Binomial Expansion?

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!

Every path in the square consists of exactly $2n$ segments; $n$ 'right' segments (R) and $n$ 'down' segments (D). Every path is a sequence of R's and D's, and the only difference between two paths is then the order of them.

For example, the $2\times2$ case has RRDD, RDRD, RDDR, DDRR, DRDR, and DRRD as the different sequences.

So the question becomes 'how many difference sequences of $n$ R's and $n$ D's can you make?' The answer is given by combinations, or the elements of Pascal's triangle. Of the $2n$ different positions, we want to choose $n$ of them to place the R's, and once that's done the sequence is completely determined. (Put the R's in the places you picked and the D's in all the others.)

Therefore, there are $\binom{2n}{n}$ different paths in an $n \times n$ grid. So for our $2 \times 2$ example, there are $\binom{4}{2} = 6$ different paths, for a $20 \times 20$ grid, there are $\binom{40}{20} = 137846528820$ paths.