Find three numbers $$a,b,c$$ between $$2$$ and $$18$$ such that
(i) their sum is $$25$$
(ii) the numbers $$2,$$ $$a, b$$ are consecutive terms of an A.P. and
(iii) the numbers $$b,c,18$$ are consecutive terms of a G.P.

The coordinates of $$A, B, C$$ are $$(6, 3), (-3, 5), (4, -2)$$ respectively, and $$P$$ is any point $$(x, y)$$. Show that the ratio of the area of the triangles $$\Delta $$ $$PBC$$ and $$\Delta $$$$ABC$$ is $$\left| {{{x + y - 2} \over 7}} \right|$$

m men and n women are to be seated in a row so that no two women sit together. If $$m > n$$, then show that the number of ways in which they can be seated is $$\,{{m!(m + 1)!} \over {(m - n + 1)!}}$$

If $${\left( {1 + ax} \right)^n} = 1 + 8x + 24{x^2} + .....$$ then $$a=..........$$ and $$n =............$$