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.

From a set of $10$ 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 most expensive ring must go at the top with the two least expensive rings adjacent to it.

Find how many three-digit numbers can be made from the digits $0,1,2,3$ and $4,$ used at most once each, if the three-digit number: three-digit number: 

must be a multiple of $10$

Find how many three-digit numbers can be made from the digits $0,1,2,3$ and $4,$ used at most once each, if the three-digit number: three-digit number: 

cannot begin with zero.