Let $$\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k$$ and $$\overrightarrow q = \widehat i + 2\widehat j + \widehat k$$ be two vectors. If a vector $$\overrightarrow r = (\alpha \widehat i + \beta \widehat j + \gamma \widehat k)$$ is perpendicular to each of the vectors ($$(\overrightarrow p + \overrightarrow q )$$ and $$(\overrightarrow p - \overrightarrow q )$$, and $$\left| {\overrightarrow r } \right| = \sqrt 3 $$, then $$\left| \alpha \right| + \left| \beta \right| + \left| \gamma \right|$$ is equal to _______________.

The ratio of the coefficient of the middle term in the expansion of (1 + x)²⁰ and the sum of the coefficients of two middle terms in expansion of (1 + x)¹⁹ is _____________.

There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of class 10 and 11 is 100 k, then k is equal to _____________.

If $$\alpha$$, $$\beta$$ are roots of the equation $${x^2} + 5(\sqrt 2 )x + 10 = 0$$, $$\alpha$$ > $$\beta$$ and $${P_n} = {\alpha ^n} - {\beta ^n}$$ for each positive integer n, then the value of $$\left( {{{{P_{17}}{P_{20}} + 5\sqrt 2 {P_{17}}{P_{19}}} \over {{P_{18}}{P_{19}} + 5\sqrt 2 P_{18}^2}}} \right)$$ is equal to _________.