A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is \underline\hspace2cm .

Permutations and Combinations question

2020 · 4 Sep · Shift 2 · Q22

2020 · 4 Sep · Shift 2 · Q22

View written solutionFree

Correct answer: 135

Each of the $6$ questions has exactly one correct answer and $3$ incorrect answers.
We want the candidate to answer all 6 questions such that exactly 4 answers are correct.
First, choose which $4$ questions are answered correctly: $\binom{6}{4} = 15$
The remaining $2$ questions must be answered incorrectly. For each such question, there are $3$ wrong choices available.

So, number of ways to answer those $2$ questions incorrectly is: $3^2 = 9$

Therefore, total number of required answer patterns is: $\binom{6}{4} \cdot 3^2 = 15 \cdot 9 = 135$
Hence, the required number of ways is: $\boxed{135}$

2020 · 4 Sep · Shift 2 · Q22

2020 · 4 Sep · Shift 2 · Q22

View written solutionFree

Correct answer: 135

Each of the $6$ questions has exactly one correct answer and $3$ incorrect answers.
We want the candidate to answer all 6 questions such that exactly 4 answers are correct.
First, choose which $4$ questions are answered correctly: $\binom{6}{4} = 15$
The remaining $2$ questions must be answered incorrectly. For each such question, there are $3$ wrong choices available.

So, number of ways to answer those $2$ questions incorrectly is: $3^2 = 9$

Therefore, total number of required answer patterns is: $\binom{6}{4} \cdot 3^2 = 15 \cdot 9 = 135$
Hence, the required number of ways is: $\boxed{135}$