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The number of ways to select 6 people out of a group of 10 such that the bride and groom must be next to each other is 8!4!4!⋅5!⋅2!
We can reduce your problem to a simpler one since you claim the bride must be in the picture. Now you have to choose 5 people out of 9.
It is simply 9!4!5!
Now that is merely selecting the participants of the picture. Since there are 6 people in the picture, you can arrange them 6! ways. Which leaves the total number of possible combinations as 9!4!5!⋅6!
I’d also like to discuss the other possible option of 6 people from a group of 10 where 2 of the 6 must be the bride and groom, and they must be beside each other. I can see this coming up in a similar situation.
This has a familiar start. We already know 2 of the 6 people, so we only need to choose 4 of the remaining 8. This is given by 8!4!4!.
Now we need to permute the group of 6 participants. One would like to say that this is simply 6! as with the first problem, but it’s a little bit different. Remember that the bride and groom must be beside each other. You can essentially think of these two people as a single person.
So the actual number of ways to permute this group of 6 people with the constraint is 5! . There’s one more stipulation. If you don’t subscribe to the notion that the groom must always be on the left (facing the couple), then you have two options to arrange our couple.
The number of ways to select 6 people out of a group of 10 such that the bride and groom must be next to each other is 8!4!4!⋅5!⋅2!
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