Binomial expansion, how to do them quickly?

Binomial expansion, how to do them quickly?

I'm currently preparing for a test where I'm bound to do a couple of
binomial expansions. Since I never encountered them in my formal
education, I looked how to do them myself and found out:
$ (x+y)^n = \displaystyle \sum_{k=0}^n \binom{n}{k} x^k y^{n-k} $
I'm using this formula for my binomial expansion, but this takes a long
time, especially to calculate the binomial coefficient (we aren't allowed
to use a calculator).
Consider the following (easy) example: Write down the binomial expansion
of $(4+x)^4$, and hence evaluate $(4.2)^2$ to 2 decimal places.
I use the above formula to get $256 + 256x + 96x^2+16x^3 + x^4$. Fill in
for $x=0.2$.
Here I encounter my first problems: How do you know how many terms to use
when you are told to evaluate to $n$ decimal places?
Unrelated to the title: $(4.2)^4 = 311.1696$. When you use 4 terms you get
$311.168$. Would the correct answer be $311.16...$ or $\approx 311.17$?
Are there other, quicker ways of doing binomial expansions, which can rid
me of the problem of binomial coefficients?