Calculating with factorials can be rough. Calculating with binomial coefficients can even be more rough. Two examples how you can counter the problems. Let’s look at two summations of binomial coefficients.

First: the sum

Let’s see

We elaborate

Let’s create an equal denominator

Simplify

or

and there we are.

Now for the sum

That is

We tackle that in the same manner with

and equalizing the denominators

so we can add the numerators:

which brings us to

Good boy! Down, down!