A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is __________

We have six different questions, and each question carries four alternative choices, exactly one of which is right.

First, we decide which questions are answered correctly. Out of the total $$6$$ questions, we want exactly $$4$$ to be correct. The number of ways of choosing these $$4$$ questions is obtained by the combination formula

$$^{n}C_{r}=\dfrac{n!}{r!\,(n-r)!}.$$

Here $$n=6$$ and $$r=4$$, so

$$^{6}C_{4}=\dfrac{6!}{4!\,2!}=15.$$

Now, for each of the chosen $$4$$ questions, there is only one way to answer correctly because only one option is the right one. Thus, the total number of answer patterns for these $$4$$ questions remains $$1^{4}=1$$.

Next, we look at the remaining $$6-4=2$$ questions, which must be answered incorrectly. Each of these questions has $$4-1=3$$ wrong alternatives. Therefore, for every question to be answered wrongly, there are $$3$$ possible choices. Since the two questions are independent, the total number of incorrect answer patterns is

$$3 \times 3 = 3^{2}=9.$$

Finally, we multiply the number of ways of choosing which questions are correct with the number of answer patterns for correct responses and with the number of answer patterns for incorrect responses:

$$\text{Total ways}=^{6}C_{4}\times 1^{4}\times 3^{2}=15\times 1\times 9=135.$$

So, the answer is $$135$$.