A box contains $$2$$ fifty paise coins, $$5$$ twenty five paise coins and a certain fixed number $$N\,\,\left( { \ge 2} \right)$$ of ten and five paise coins. Five coins are taken out of the box at random. Find the probability that the total value of these $$5$$ coins is less than one rupee and fifty paise.

Let $$OA$$ $$CB$$ be a parallelogram with $$O$$ at the origin and $$OC$$ a diagonal. Let $$D$$ be the midpoint of $$OA.$$ Using vector methods prove that $$BD$$ and $$CO$$ intersect in the same ratio. Determine this ratio.

Evaluate $$\int\limits_0^1 {\log \left[ {\sqrt {1 - x} + \sqrt {1 + x} } \right]dx} $$

Sum of the first n terms of the series $${1 \over 2} + {3 \over 4} + {7 \over 8} + {{15} \over {16}} + ............$$ is equal to