1. Let us state the problem: We have a multiple choice test with 20 questions, each having 4 possible answers (a,b,c,d), only one of which is correct. 2. We want the probability of getting exactly 8 questions correct by guessing. 3. Since each question has 4 answers, the probability of getting a question correct by guessing is $p = \frac{1}{4}$. 4. The probability of getting a question wrong is $q = 1 - p = \frac{3}{4}$. 5. The number of trials is $n=20$, and we want exactly $k=8$ successes. 6. The binomial probability formula is: $$ P(X=k) = \binom{n}{k} p^k q^{n-k} $$ 7. Calculate the binomial coefficient: $$ \binom{20}{8} = \frac{20!}{8! \cdot 12!} $$ 8. Substitute values: $$ P(X=8) = \binom{20}{8} \left(\frac{1}{4}\right)^8 \left(\frac{3}{4}\right)^{12} $$ 9. This can be computed using a calculator or software for exact decimal value. Final answer: $$ P(X=8) = \binom{20}{8} \left(\frac{1}{4}\right)^8 \left(\frac{3}{4}\right)^{12} $$