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 create an equal denominator
and there we are.
Now for the sum
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!