### Instruction: 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?### Response: I want to count the number of ways to choose $3$, $4$, ..., or $10$ questions to be false, and then assign true to the remaining questions.
For each number $k$ from $3$ to $10$, I can use the binomial coefficient $\binom{10}{k}$ to count the ways to select $k$ questions out of $10$ to be false.
Then, I can add up all these binomial coefficients to get the total number of ways.
Alternatively, I can use the complement principle and subtract the number of ways to answer the test with $0$, $1$, or $2$ falses from the total number of ways to answer the test, which is $2^{10}$.
This might be faster, since I only have to calculate $3$ binomial coefficients instead of $8$.