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!
Best Answer
HINT: Label each intersection with the number of ways in which Joan can reach it. She can reach her house in exactly one way: she takes no steps. She can reach the intersection a block to the east in one way: she must take one eastward step. Similarly, she can reach the intersection a block to the north in one way: she must take one northward step. She can reach the intersection one block to the northeast of her house, however, in two ways: she can go east, then north, or she can go north, then east.
I’ve started the labelling process in the grid below, whose cells correspond to the street intersections, not to the city blocks; I’ve included two more labels besides the ones mentioned above. Note that in general, she can reach each intersection either from the one immediately south of it or from the one immediately west of it, unless it’s on the southern or western boundary of the square.
$$\begin{array}{|c|c|c|c|c|} \hline ?&?&?&?&?&?\\ \hline ?&?&?&?&?&?\\ \hline ?&?&?&?&?&?\\ \hline 1&3&?&?&?&?\\ \hline 1&2&3&?&?&?\\ \hline 1&1&1&?&?&?\\ \hline \end{array}$$
If you finish filling in the labels correctly, both the labels themselves and the way in which you calculated them should give you the connection with Pascal’s triangle.