If ${ }^{27} P_{r+7}=7722{ }^{25} P_{(r+4)}$, then $r=$

If the number of diagonals of a regular polygon is 35 , then the number of sides of the polygon is

If four letters are chosen from the letters of the word ASSIGNMENT and are arranged in all possible ways to form 4 letter words (with or without meaning), then total number of such words that can be formed is

The terms containing $x^r y^s$ (for certain $r$ and $s$ ) are present in both the expansions of $\left(x+y^2\right)^{13}$ and $\left(x^2+y\right)^{14}$. If $\alpha$ is the number of such terms, then the $\operatorname{sum} \alpha \sum_{r, s}(r+s)=$