SATHEE: Permutations and Combinations 2 Question 17

Permutations and Combinations 2 Question 17

17. In a certain test, $a_{i}$ students gave wrong answers to at least $i$ questions, where $i = 1, 2, \dots, k$ . No student gave more that $k$ wrong answers. The total number of wrong answers given is … .

$(1982, 2 M)$

True/False

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Answer:

Correct Answer: 17. $2^{n} - 1$

Solution:

The number of students answering exactly $k (1 \leq k \leq n - 1)$ questions wrongly is $2^{n - k} - 2^{n - k - 1}$ . The number of students answering all questions wrongly is $2^{0}$ .

Thus, total number of wrong answers

$\begin{matrix} = 1 (2^{n - 1} - 2^{n - 2}) + 2 (2^{n - 2} - 2^{n - 3}) + \dots \\ + (n - 1) (2^{1} - 2^{0}) + 2^{0} \cdot n \\ = 2^{n - 1} + 2^{n - 2} + 2^{n - 3} + \dots + 2^{1} + 2^{0} = 2^{n} - 1 \end{matrix}$

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Permutations and Combinations 2 Question 17

17. In a certain test, ai students gave wrong answers to at least i questions, where i=1,2,…,k. No student gave more that k wrong answers. The total number of wrong answers given is … .

True/False

Answer:

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17. In a certain test, $a_{i}$ students gave wrong answers to at least $i$ questions, where $i = 1, 2, \dots, k$ . No student gave more that $k$ wrong answers. The total number of wrong answers given is … .