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 $P(k;n,p)=C(n,k)\ast p^k\ast q^{(n-k)}$, 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 $P(6;10,\frac{1}{4})=C(10,6)\ast (\frac{1}{4}{)}^6\ast (\frac{3}{4}{)}^{(10-6)}$

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

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

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