A test has $6$ questions with $4$ possible answers for each (a,b,c,d), plus $5$ more true or false questions.
How many students are required to take the test to guarantee that $2$ write down the identical answers?
I was thinking it might be something like... $$6C4\times 5C2=150$$
Is this the right line of thinking? Any help would be much appreciated!
On
The number of ways the questions can be answered (assuming all are answered) is found as follows: The first 6 questions can be answered in 4 ways each, for $4^6$ ways in all. The 5 true-false questions can be answered in $2^5$ ways. So the number of different answer sheets for the exam is $4^6\cdot 2^5$. To guarantee two identical solutions, $4^6\cdot 2^5+1$ students are required.
How many ways can you answer the questions?
Think of it as flipping a coin; if you flip a coin $3$ times there are $2^3$ possibilities.
Therefore there are $$4^6\times2^5=2^{17}$$ ways of answering your test.
In the worst case scenario, all $2^{17}$ students all have different answers. So when $1$ extra student is added, they must get the same answer to another student.
Thus we have $$2^{17}+1$$ students needed to ensure $2$ identical answers.