The number of 6-digit numbers made by using the digits 1, 2, 3, 4, 5, 6, 7, without repetition and which are multiple of 15 is ____________.

Suppose $$\mathop {\lim }\limits_{x \to 0} {{F(x)} \over {{x^3}}}$$ exists and is equal to L, where

$$F(x) = \left| {\matrix{ {a + \sin {x \over 2}} & { - b\cos x} & 0 \cr { - b\cos x} & 0 & {a + \sin {x \over 2}} \cr 0 & {a + \sin {x \over 2}} & { - b\cos x} \cr } } \right|$$.

Then, $$-$$112 L is equal to ___________.

If for some $$\alpha$$ > 0, the area of the region $$\{ (x,y):|x + \alpha | \le y \le 2 - |x|\} $$ is equal to $${3 \over 2}$$, then the area of the region $$\{ (x,y):0 \le y \le x + 2\alpha ,\,|x| \le 1\} $$ is equal to ____________.

Let $$f(t) = \int\limits_0^t {{e^{{x^3}}}\left( {{{{x^8}} \over {{{({x^6} + 2{x^3} + 2)}^2}}}} \right)dx} $$. If $$f(1) + f'(1) = \alpha e - {1 \over 6}$$, then the value of 150$$\alpha$$ is equal to ___________.