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)?

(a) 725760 ways

(b) 3265920 ways

(c) 2903040 ways

Explanation:

Part (a):

As there are 10 people in total so there will be total ways to line up = 10!

Now it is the condition that bride and groom must be next to each other so we will count them as one entity, the remaining ways will be = 9!

As the bride and groom can swap there places so the number of ways will become = 9! * 2

Number of ways = 362880 * 2 = 725760 ways

Part (b):

Now set the groom to leftmost position: 9! (as one person is fixed)

When groom is not in left most position = 10! - 9! = 3265920 ways

Part (c):

when the groom is not at end, this means neither he should be on right nor on left so, this could be calculated as = 10! - 2* 9! = 2903040 ways

i hope it will help you!