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  2¹⁰  ways that the 10-question test  =  1024 ways.

 

To find how many ways at least 3 of the questions have been answered false, it will be easier to calculate how many times that isn't done and subtract that from the total number of ways.

 

To answer no questions false, there is only 1 way.

To answer one questions false, there are 10 ways.

To answer two questions false, there are 45 ways.    [  ₁₀C₂   =  45 ]

 

So, to answer fewer than 3 questions false, there are 1 + 10 + 45  =  56 ways.

 

Subtracting, there are 1024 - 56  =  968 ways to answer at least 3 questions false.