The last bullet point is usually taken as the definition of \({n \choose k}\text{.}\) Out of \(n\) objects we must choose \(k\) of them, so there are \(n\) choose \(k\) ways of doing this. Each of our counting problems above can be viewed in this way:
How many subsets of \(\{1,2,3,4,5\}\) contain exactly 3 elements? We must choose \(3\) of the 5 elements to be in our subset. There are \({5 \choose 3}\) ways to do this, so there are \({5 \choose 3}\) such subsets.
How many bit strings have length 5 and weight 3? We must choose \(3\) of the 5 bits to be 1's. There are \({5 \choose 3}\) ways to do this, so there are \({5 \choose 3}\) such bit strings.
How many lattice paths are there from (0,0) to (3,2)? We must choose 3 of the 5 steps to be towards the right. There are \({5 \choose 3}\) ways to do this, so there are \({5 \choose 3}\) such lattice paths.
What is the coefficient of \(x^3y^2\) in the expansion of \((x+y)^5\text{?}\) We must choose 3 of the 5 copies of the binomial to contribute an \(x\text{.}\) There are \({5 \choose 3}\) ways to do this, so the coefficient is \({5 \choose 3}\text{.}\)