Find the number of ways in which 12 children can be divided into 2 groups of 6 if two particular boys must be in different groups.

The number of ways of choosing 6 boys from the 12 to form
the two groups of 6 boys is given by C(12,6)

In the case
where the two boys are in one group, the number of ways of forming the groups is C(10,
4). This is due to the fact that 2 boys are taken together for a group, so now we need
to choose another 4 boys from the 10 remaining to complete the
group.

So the number of ways of forming the groups where
the two boys are in different groups can be calculated by subtracting the case where
they are together from the general case: this is given by C(12, 6) - C(10 ,
6)

= 12! / (6!* 6!) - 10! /
(6!*4!)

= 924 - 210

=
714

Therefore the number of ways the children
can be divided is 714.