Question

In a certain test, there are $n$ questions. In this test $(n-r{)}^2$ students gave wrong answers to at least $r$ questions ($1\le r\le n$). If total number of wrong answers given is $204$, then the value of $n$ is

Answer 1

no. of students giving wrong answers to at least $r$ questions $=(n-r{)}^2$
no. of students giving wrong answers to exactly $r$ questions $=(n-r{)}^2-(n-r-1{)}^2=2n-2r-1$

Total no. of wrong answers $=\sum _{r=1}^{n-1}(2n-2r-1)\cdot r=(2n-1)\sum _{r=1}^{n-1}r-2\sum _{r=1}^{n-1}{r}^2=\frac{(2n-1)(n-1)\left(n\right)}{2}-2\cdot \frac{n(n-1)(2n-1)}{6}\Rightarrow \frac{n(n-1)(2n-1)}{6}=204\Rightarrow n=9$