KöMaL Survey on a total of 14 questions, with 4 possible answers for each question. The best result was 10 correct answers. What is the probability that we achieve this result by random filling out?
I. solution. In total, there are $4^{14}$ possible ways to fill out the answers. 10 correct answers occur when 4 answers are wrong. There are $\binom{14}{4}=1001$ ways to choose which 4 answers are wrong. For each of these, we can give 3 incorrect answers, which results in $1001 \cdot 3^{4}=81081$ possible fill-ins.

Thus, the probability is: $\frac{81081}{4^{14}}=0.00030205$.

II. solution. The probability of giving a correct answer to a given question is 0.25, and the probability of giving a wrong answer is 0.75.

There are $\binom{14}{10}$ ways to choose which 10 questions we answer correctly out of the 14 questions. For the remaining 4 questions, we give incorrect answers. Therefore, the sought probability is:

$$
\binom{14}{10} \cdot 0.25^{10} \cdot 0.75^{4}=3.0205 \cdot 10^{-4}
$$