Permutations and Combinations 2 Question 17
17. In a certain test, $a _i$ students gave wrong answers to at least $i$ questions, where $i=1,2, \ldots, k$. 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
$$ \begin{gathered} =1\left(2^{n-1}-2^{n-2}\right)+2\left(2^{n-2}-2^{n-3}\right)+\ldots \\ +(n-1)\left(2^{1}-2^{0}\right)+2^{0} \cdot n \\ =2^{n-1}+2^{n-2}+2^{n-3}+\ldots+2^{1}+2^{0}=2^{n}-1 \end{gathered} $$
