At a wedding reception, the bride and groom and eight attendants will form a receiving line. How many ways can they be arranged in each of following cases?
a) Any order will do.
b) The bride and groom must be the last two in line.
c) The groom must be last in line with the bride next to him.
a) In how many ways can the receiving line be formed if any order will do?
3628800
b) In how many ways can the receiving line be formed if the bride and groom must be the last two in line?
Answer
For part b), we consider the bride and groom as a single entity, since they must be together at the end of the line. This means we now have 9 entities to arrange (8 attendants + 1 bride/groom pair), which can be done in 9! ways. However, the bride and groom can switch places with each other, so we need to multiply by 2 to account for these two possibilities. So, the number of ways the receiving line can be formed if the bride and groom must be the last two in line is \(\boxed{725760}\).
Step 1 ：For part a), we have 10 people who can be arranged in any order. This is a simple permutation problem, where the number of arrangements is given by 10!. So, the number of ways the receiving line can be formed if any order will do is \(\boxed{3628800}\).
Step 2 ：For part b), we consider the bride and groom as a single entity, since they must be together at the end of the line. This means we now have 9 entities to arrange (8 attendants + 1 bride/groom pair), which can be done in 9! ways. However, the bride and groom can switch places with each other, so we need to multiply by 2 to account for these two possibilities. So, the number of ways the receiving line can be formed if the bride and groom must be the last two in line is \(\boxed{725760}\).
