A multiple choice exam has 10 questions. Each question has four possible answers, of which only one is correct. If a student guesses all the answers, what is the probability that the student will answer exactly 6 questions correctly?

Step 1 ：Given that the total number of questions (n) is 10, the number of successful outcomes (k) is 6, the probability of success (p) is \(\frac{1}{4}\), and the probability of failure (q) is \(\frac{3}{4}\). We can use the binomial probability formula to find the probability of exactly 6 correct answers.

Step 2 ：The binomial probability formula is \(P(k;n,p) = C(n, k) * p^k * q^{(n-k)}\), where \(C(n, k)\) is the number of combinations of n items taken k at a time

Step 3 ：Substituting the given values into the formula, we get \(P(6;10,\frac{1}{4}) = C(10, 6) * (\frac{1}{4})^6 * (\frac{3}{4})^{(10-6)}\)

Step 4 ：Calculating the combination \(C(10, 6)\), we get 210

Step 5 ：Substituting this value into the equation, we get \(P(6;10,\frac{1}{4}) = 210 * (\frac{1}{4})^6 * (\frac{3}{4})^{4}\)

Step 6 ：Calculating the above expression, we get the required probability