The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :

If $$\overrightarrow a = \widehat i + 2\widehat k,\overrightarrow b = \widehat i + \widehat j + \widehat k,\overrightarrow c = 7\widehat i - 3\widehat j + 4\widehat k,\overrightarrow r \times \overrightarrow b + \overrightarrow b \times \overrightarrow c = \overrightarrow 0 $$ and $$\overrightarrow r \,.\,\overrightarrow a = 0$$. Then $$\overrightarrow r \,.\,\overrightarrow c $$ is equal to :

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is :

Let $$\mathrm{S} = \{ {w_1},{w_2},......\} $$ be the sample space associated to a random experiment. Let $$P({w_n}) = {{P({w_{n - 1}})} \over 2},n \ge 2$$. Let $$A = \{ 2k + 3l:k,l \in N\} $$ and $$B = \{ {w_n}:n \in A\} $$. Then P(B) is equal to :