In how many ways can a photographer at a wedding arrange $6$ people in a row, including the bride and the groom, if the bride should be positioned in the groom's left.
Don't have a idea to start this one.
[Math] Photographer arranging six people in a row
combinatorics
Best Answer
Since the bride has to be on the left of the groom there are $5$ admissible positions for the groom (second, third, etc, sixth but not first). If the position of the groom is $k$ for $k=2,3,4,5,6$ then you can place the bride in $k-1$ positions that are on the left. This gives you $$\sum_{k=2}^{6}(k-1)=\sum_{k=1}^5k=\frac{5(5+1)}{2}=15$$ ways to place them. In each of these ways, the rest of the positions may be filled in an arbitrary way, so there are $4!$ ways to do that, which gives a total of $$15\cdot4!=360$$
Of course the simplest answer is mentioned in the comments: Due to symmetry there are as many ways to place the bride on the left of the groom as to place her on the right of the groom. Since the total ways are $6!$ this gives immediately that the answer is $$\frac{6!}{2}=\frac{720}{2}=360$$ thus confirming the above approach.