Each letter $A$ to $Z$ is encrypted (or transformed) to a fixed distinct letter using its position in the alphabet $(\mathrm{A}=1, \mathrm{B}=2,\mathrm{C}=3,\dots ).$ By doing this, the password $\mathrm{SATURN}$ is encrypted as $\mathrm{ECHKBP}$. Similarly the encrypted password for a word is $\mathrm{UJSNOL}$. Find the number of possibilities for the original word.

Without using a calculator, find the value of:

$\frac{5!}{3!}$

Without using a calculator, find the value of:

$\frac{4!}{2!}-3!$

Without using a calculator, find the value of:

$7\times 4!+21\times 3!$

Two children are selected at random from a group of six boys and four girls. Use combinations to find the probability of selecting:

two girls

Three chocolates are selected at random from a box containing $10$ milk chocolates and $15$ dark chocolates. Find the probability of selecting exactly:

two milk chocolates

A farmer has $50$ animals. They have $24$ sheep, of which three are male, and they have $26$ cattle, of which $20$ are female. A veterinary surgeon wishes to test six randomly selected animals. Find the probability that the selection consists of:

equal numbers of cattle and sheep