Question

If ${\sin }^2{\theta }_1+{\sin }^2{\theta }_2+\cdot \cdot \cdot +{\sin }^2{\theta }_{104}=0$ where ${\theta }_i\in [0,\pi ],i=1,2,...,104$, then the different sets of values of $({\theta }_1,{\theta }_2,{\theta }_3,\cdots ,{\theta }_{104})$ for which $\cos {\theta }_1+\cos {\theta }_2+\cdots +\cos {\theta }_{104}=100$ is

Answer 1

The given expression: ${\sin }^2{\theta }_1+{\sin }^2{\theta }_2+\cdot \cdot \cdot +{\sin }^2{\theta }_n=0$ is possible only if $\sin {\theta }_1=\sin {\theta }_2=\cdots =\sin {\theta }_n=0$
$\Rightarrow \theta =0$ or $\pi$ $(\because \theta \in [0,\pi ])$

Now,
$\cos {\theta }_1+\cos {\theta }_2+\cdots +\cos {\theta }_n=n-4$ that implies any two of ${\theta }_1,{\theta }_2,\cdots ,{\theta }_{104}$ must be $\pi$ and rest should be $0$.
So, total number of ways is ${}^{104}{C}_2=5356$