Test 10 questions

A test has 10 questions with multiple-choice answers. For the choice of the correct answer there are four options A, B, C, D, where always one of them is correct. How many are there all different possibilities for answers in this test,
a) if its taker answers every question
b) if he answers every question and correctly answers exactly 2 questions.
c) he answers every question and correctly answers at most 2 questions
d) he answers every question and correctly answers at least 9 questions.

Final Answer:

a =  1048576
b =  295245
c =  885735
d =  31

Step-by-step explanation:

$a=4^{10}=1048576$

$$\begin{gathered} C_2(10)=\left(\genfrac{}{}{0ex}{}{10}{2}\right)=\frac{10!}{2!(10-2)!}=\frac{10\cdot 9}{2\cdot 1}=45 \\ b=\left(\genfrac{}{}{0ex}{}{10}{2}\right)\cdot (4-1{)}^{10-2}\cdot 1^2=45\cdot (4-1{)}^{10-2}\cdot 1^2=295245 \end{gathered}$$

$$\begin{gathered} C_1(10)=\left(\genfrac{}{}{0ex}{}{10}{1}\right)=\frac{10!}{1!(10-1)!}=\frac{10}{1}=10 \\ c=b+\left(\genfrac{}{}{0ex}{}{10}{1}\right)\cdot (4-1{)}^{10-1}\cdot 1^1\cdot 3^1=295245+10\cdot (4-1{)}^{10-1}\cdot 1^1\cdot 3^1=885735 \end{gathered}$$

$d=\left(\genfrac{}{}{0ex}{}{10}{1}\right)\cdot 1^9\cdot 3^1+1\cdot 1^{10}=10\cdot 1^9\cdot 3^1+1\cdot 1^{10}=31$

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