How many ways are there to answer a $10$-question true/false test, where at least $3$ of the questions have been answered with a false?
There are $2^{10} = 1024$ ways to answer the questions on the true/false test. However, we can also compute the number of ways of answering the questions on the true/false test by using combinations; there are $\binom{10}{k}$ ways to answer $k$ of the questions on the test with a false, so we have: $$\binom{10}{0} + \binom{10}{1} + \cdots + \binom{10}{9} + \binom{10}{10} = 2^{10}.$$ The desired answer is \begin{align*}
&\binom{10}{3} + \binom{10}{4} + \cdots + \binom{10}{9} + \binom{10}{10} \\
=\text{ }&2^{10} - \binom{10}{0} - \binom{10}{1} - \binom{10}{2} = 1024 - 1 - 10 - 45 \\
=\text{ }&\boxed{968}.
\end{align*}