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

Answer

`2^(n) - 1`

Explanation

The number of students answering exactly `k(1 le k le 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
`= 1(2^(n- 1) - 2^(n - 2)) + 2(2^(n - 2) - 2^(n - 3)) + ... + (n - 1)(2^(1) - 2^(0)) + 2^(0) * n`
`= 2^(n- 1) + 2^(n - 2) + 2^(n - 3) + ... + 2^(1) - 2^(0) = 2^(n)-1`