20 Multiple Choice Questions but can leave 2 blank? Counting There are 20 MCQ consisting of four answers each, in how many ways can a student answer if the student can leave two blank answers
I already know that if there are 20 MCQ but can leave one blank then its $(5\times 4^{19}) \times 20$, but what changes if I can also add one extra blank?
 A: The answer you mention is not correct. I imagine it comes from saying "choose the question you may leave blank ($20$ options), then there are $5$ choices for that question and $4$ each for the other $19$ questions", but this argument is incorrect, since it counts all the ways of not leaving a question blank $20$ times each (because which question you chose not to leave blank doesn't matter).
The correct answer for leaving at most one blank is $4^{20}+\binom{20}1\times4^{19}$: the first term is the number of ways to leave none blank, and the second the number of ways to leave exactly one blank. Can you see how to extend this to include possibly leaving two blank?
A: Hint:
If there is exactly one blank answer then there are $20$ possibilities for the question that is answered with a blank, and for each of the remaining questions there are $4$ possible answers.
That gives $20\cdot4^{19}$ possibilities in total

Now likewise try to find the number of possibilities if there are exactly $2$ blank answers.
Final result is addition of possibilities with exactly $0$, exactly $1$ and exactly $2$ blank answers.
A: If the student will leave $2$ blank it is $${20\choose{18}} \times 4^{18} \approx 1.3 \times 10^{13}$$ but if a student may leave $2$,$1$ or no blanks it is $$ {20\choose{18}} \times 4^{18} + {20\choose{19}} \times 4^{19} + 4^{20} \approx 1.97\times 10^{13}$$
