Three distinct numbers are selected from first $100$ natural numbers. The probability that all the three numbers are divisible by $2$ and $3$ is

The number of $3 \times 3$ matrices $A$ whose entries are either $0$ or $1$ and for which the system $\mathrm{A}\left[\begin{array}{l}\mathrm{x} \\ \mathrm{y} \\ \mathrm{z}\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has exactly two distinct solutions, is

If four persons are chosen at random from a group of $3$ men, $2$ women and $4 $ children. Then the probability that exactly two of them are children, is

Six boys and six girls sit in a row randomly. The probability that the six girls sit together

A mapping is selected at random from the set of all the mappings of the set $A = \left\{ {1,\,\,2,\,...,\,n} \right\}$ into itself. The probability that the mapping selected is an injection is

Three mangoes and three apples are in a box. If two fruits are chosen at random, the probability that one is a mango and the other is an apple is