I'm working on some permutation problems for my discrete math course and finding that I'm not fully understanding the material. I've watched a couple YouTube videos and read a couple different math books, and it's still not fully solidifying. I was hoping I could post my problems and that someone could help me think through the problems so that I may better understand the material. Here are the current problems and my current answers:
Ten members of a wedding party are lining up in a row for a photograph.
Problem #1:
How many ways are there to line up the ten people?
Answer for Proble #1:
$10! = 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 3,628,800$
Problem #2:
How many ways are there to line up the ten people if the groom must be to the immediate left of the bride in the photo?
Answer for Question #2:
$9! = 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 362,880$
Problem #3:
How many ways are there to line up the ten people if the bride must be next to the maid of honor and the groom must be next to the best man?
Answer for Question #3:
We can group together the bride and maid-of-honor into one unit with 2 order possibilities: {(bride,maid-of-honor), (maid-of-honor, bride)}, and we can also group together the groom and best main into one unit with 2 order possibilities: {(groom, best man), (best man, groom)}.
Essentially, since these units are grouped together and must stay with one another when ordering, instead of 10 spots in the arrangement, we can think of there actually being 8 spots. For each permutation, there will be 3 extra permutations since the members of each group can swap places. For instance, labeling each member as a number and placing the grouped units at the end of the lineup we get 4 different permutations:
Ultimately, we have the following total permutations:
$8! \cdot 4 = \left (8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \right ) \cdot 4 = 161,280$
Any help is greatly appreciated. Thank you all so much! I hope you all are doing well and staying safe.