jmc

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 $\left(\genfrac{}{}{0ex}{}{10}{k}\right)$ ways to answer $k$ of the questions on the test with a false, so we have: $$\left(\genfrac{}{}{0ex}{}{10}{0}\right)+\left(\genfrac{}{}{0ex}{}{10}{1}\right)+\cdots +\left(\genfrac{}{}{0ex}{}{10}{9}\right)+\left(\genfrac{}{}{0ex}{}{10}{10}\right)=2^{10}.$$ The desired answer is $$\begin{array}{rl}& \left(\genfrac{}{}{0ex}{}{10}{3}\right)+\left(\genfrac{}{}{0ex}{}{10}{4}\right)+\cdots +\left(\genfrac{}{}{0ex}{}{10}{9}\right)+\left(\genfrac{}{}{0ex}{}{10}{10}\right)\\ =& 2^{10}-\left(\genfrac{}{}{0ex}{}{10}{0}\right)-\left(\genfrac{}{}{0ex}{}{10}{1}\right)-\left(\genfrac{}{}{0ex}{}{10}{2}\right)=1024-1-10-45\\ =& \boxed{968}.\end{array}$$