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**Topic**

In a multiple-choice test with 50 questions, each question has 4 answer options A, B, C, D. Each correct answer is added 0.2 points and each wrong answer is deducted. go 0.1 point. If a candidate randomly selects the answers to all 50 questions, what is the probability of getting 9.4 points on the above test?

**Solution method – See details**

Step 1: Set hidden *x* is the number of correct answers, represent the number of wrong answers in terms of *x*

Step 2: Express the number of points achieved in *x* get an equation whose right side is 9.4

Step 3: Solve the equation found in step 2 to find *x*

Step 4: With the number of correct/false answers known to get 9.4 points, find the number of ways to choose *x* correct sentence and the number of ways to choose the right/wrong option to find the number of possibilities

**Detailed explanation**

Call *x* is the number of correct answers (*x* > 0)

So 50 – *x* is the number of wrong answers

Points are awarded for correct answers *x* sentence is: 0.2.*x*

Points deducted for incorrect answers 50 – *x* the sentence is: 0.1.(50 – *x*)

We have a candidate score of 9.4

Deduce 0.2.*x *– 0.1.(50 – *x*) = 9.4 \( \Leftrightarrow 0.2x – 5 + 0.1x = 9.4 \Leftrightarrow 0.3x = 14.4 \Leftrightarrow x = 48\)

Therefore, candidates who get 48 questions correct and 2 questions wrong get 9.4 points.

Number of ways to choose 48 correct answers in 50 questions of the exam, there are \(C_{50}^{48}\) choices

In each question, the number of ways to choose 1 correct answer is: 1 way

In each question, the number of ways to choose 1 wrong answer out of 3 wrong options is: 3 ways

Since each question has 1 right answer and 3 wrong options, the number of possibilities of getting 9.4 points in the above test is: \(C_{50}^{48}{.1.3^2} = 11025\)

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