JEE Main 2022 · 24 Jun · Shift 2 · Mathematics Q28

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MathematicsInteger+4 / -1Question 28 of 90

In an examination, there are 10 true-false type questions. Out of 10, a student can guess the answer of 4 questions correctly with probability $\dfrac34$ and the remaining 6 questions correctly with probability $\dfrac14$ . If the probability that the student guesses the answers of exactly 8 questions correctly out of 10 is $\dfrac{27k}{4^{10}}$ , then $k$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 479

Split 8 correct as $i$ from the four ( $p=\frac34$ ) and $j$ from the six ( $p=\frac14$ ): $(4,4)$ , $(3,5)$ , $(2,6)$ .

Numerators over $4^{10}$ .

$(4,4)$ : $3^4\cdot\binom64\cdot3^2=81\cdot135=10935$ .
$(3,5)$ : $\binom43\,3^3\cdot\binom65\,3=4\cdot27\cdot6\cdot3=1944$ .
$(2,6)$ : $\binom42\,3^2\cdot1=54$ .

Total. $12933=27k$ , so $k=479$ .

Answer: 479

Topics:Binomial distribution Independent events

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JEE Main 2022 · 24 Jun · Shift 2, Mathematics question 28