The binomial theorem is a useful formula that helps us expand binomials raised to a positive integer power (i.e. $(a+b)^n$). Here it is:
$$ (a+b)^n = \sum_{k=0}^n{n\choose k}a^{n-k}b^k $$
Here, $n \choose k$ (pronounced “$n$ choose $k$”) is defined as follows:
$$ {n \choose k} = \frac{n!}{k!(n-k)!} $$
$n!$ (pronounced $n$ factorial) is the product $1 \times 2 \times 3 \times \cdots \times n$. The exception is if $n=0$: the factorial of 0 is defined as 1.
Note: $n \choose k$ tells you how many unique ways there are to choose $k$ items from a set of $n$ items, assuming order doesn’t matter. It’s also the value of the $(k+1)$th number on the $n$th row of Pascal’s triangle (assuming the top is row 0).
You can use the binomial theorem to derive some interesting summations. I’m going to let you do these yourself because these are homework problems and I don’t want to give out the answers directly.