In how many ways can gold, silver and bronze medals be awarded for first, second and third places in a race between $20$ athletes? You may assume that no two athletes tie in these positions.;

Find how many of the arrangements of four letters from $A,B,C,D,E$ and $F:$

begin with the letter $A$

Find how many of the arrangements of four letters from $A,B,C,D,E$ and $F:$

contain the letter $A.$

From a group of $10$ boys and seven girls, two are to be chosen to act as the hero and the villain in the school play. Find in how many ways this can be done if these two roles are to be played by any of the children.

From a group of $10$ boys and seven girls, two are to be chosen to act as the hero and the villain in the school play. Find in how many ways this can be done if these two roles are to be played by two girls or two boys.

From a group of $10$ boys and seven girls, two are to be chosen to act as the hero and the villain in the school play. Find in how many ways this can be done if these two roles are to be played by a boy and a girl.

From a set of $10$ distinct rings, a jeweller wishes to display seven of them in their shop window. The formation of the display is shown in the diagram opposite. Find the number of possible displays if, from the set of $10 :$

the ring with the largest diamond must go at the top of the display.