(a) Considering the girls as one unit, and the six boys make up 7. These 7 can be seated in 7! ways. The girls among themselves may be relatively interchanged in 6! ways.
The number of arrangements is 7! .6! = 3628800
(b) In this case first arrange the six boys in 6! ways. Suppose,
is one such arrangement.
Then the six girls can have their positions between any two of these boys, the arrangement starting with a girl.
and the number of arrangements is (!6)2 (or) the positions of the six girls can be (arrangement now starting with a boy)
and the number of arrangements of six boys and six girls (seated alternately) is 2(6!)2 = 1036800.
(c) In this case, the girls are separated, not necessarily by only one boy, between any two girls. First the boys are arranged in 6! ways.
The positions available for the six girls can be chosen from the seven (as indicated). The girls are arranged in 7 6 P 6! ways. Therefore, the total number of arrangement is 6! 7! = 3628800
In the case of circular permutation, the arrangements, correspondingly, are
(i) 6! 6! = 518400
(ii) 5! 6! = 86400
(iii) 5! 6! = 86400
Note: In this case, whether they sit alternately, or one group (of boys) is separated by the other (of girls) the effect is the same. Hence, in (ii) and (iii) cases, number of arrangements are equal.