Given the vectors:
\[ \mathbf{a} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} \]
\[ \mathbf{b} = 3(\mathbf{i} - \mathbf{j} + \mathbf{k}) = 3\mathbf{i} - 3\mathbf{j} + 3\mathbf{k} \]
where
\[ \mathbf{a} \times \mathbf{c} = \mathbf{b} \]
\[ \mathbf{a} \cdot \mathbf{x} = 3 \]
Find:
\[ \mathbf{a} \cdot (\mathbf{x} \times \mathbf{b} - \mathbf{c}) \]
If three numbers are randomly selected from the set \( \{1,2,3,\dots,50\} \), then the probability that they are in arithmetic progression is:
A student has to write the words ABILITY, PROBABILITY, FACILITY, MOBILITY. He wrote one word and erased all the letters in it except two consecutive letters. If 'LI' is left after erasing then the probability that the boy wrote the word PROBABILITY is: \