Question 1: You take a statistics exam. The test is a multiple choice questionnaire comprising 7 independent questions. Each question has 9 possible answers, 3 of which are correct. A student is considered to have answered a question correctly if and only if all 3 correct answers have been selected. You will pass the exam if you answer at least 4 questions correctly. We assume that for each question you randomly choose 3 answers. You answer the first question of the exam by randomly choosing 3 answers. What is the probability that you chose all 3 correct answers?

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Answer to a math question Question 1: You take a statistics exam. The test is a multiple choice questionnaire comprising 7 independent questions. Each question has 9 possible answers, 3 of which are correct. A student is considered to have answered a question correctly if and only if all 3 correct answers have been selected. You will pass the exam if you answer at least 4 questions correctly. We assume that for each question you randomly choose 3 answers. You answer the first question of the exam by randomly choosing 3 answers. What is the probability that you chose all 3 correct answers?

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Let's calculate the probability of choosing all 3 correct answers on the first question:

The total number of ways to choose 3 answers from 9 possible answers is given by the combination formula:

\text{Total possible ways}=\binom{9}{3}=\frac{9!}{3!\left(9-3\right)!}

The number of ways to choose all 3 correct answers is 1, as there is only one correct combination of 3 answers out of the 9:

\text{Number of correct ways} = 1

Therefore, the probability of choosing all 3 correct answers on the first question is:

\text{Probability}=\dfrac{\text{Number of correct ways}}{\text{Total possible ways}}=\dfrac{1}{\frac{9!}{3!\left(9-3\right)!}}

\text{Probability}=\dfrac{1}{\frac{9!}{3!\left(9-3\right)!}}=\dfrac{1}{84}

\boxed{\text{Probability} = \dfrac{1}{84}}

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