Lining up a wedding party for a photo.

1) Ten members of a wedding party are lining up in a row for a photograph.

(a)How many ways are there for the wedding party to line up in which the bride is not next to the groom?
(b)How many ways are there for the wedding party to line up if the groom is not in the leftmost position?
(c)How many ways are there for the wedding party to line up if the groom is not at one end (i.e. in the leftmost or rightmost positions)?

Step-by-step explanation:

a) Number of arrangements in which bride is not next to the groom

= Total number of arrangements - Number of arrangements in which bride and groom are next to each other

= 10! - 2*9!

= 2903040

b) Number of arrangements such that groom is not at the leftmost position

= Total number of arrangements - Number of arrangements with groom at leftmost position

= 10! - 9!

= 3265920

c) Number of ways such that groom is not at one end

= Total number of arrangements - Number of arrangements with groom at one end

= 10! - 2*9!

= 2903040