7. Recall that the number of ways to arrange n items in a row is n!.
a) Think of the bride and groom as one item, which means there are five items to be arranged, giving 5! arrangements. However, the bride and groom can be arranged in two different ways, so for each of the 5! arrangements, there are two possibilities for the arrangement of the bride and groom. The total number of permutations in which the bride is next to the groom is 2 · 5! = 240.
b) It is easiest to calculate the number of ways for the groom's mother to be next to the bride and then take this number away from the total number of ways to arrange the six people. As in part a), there are 240 permutations in which the groom's mother is next to the bride. There is a total of 6! = 720 arrangements of the six people. Hence the total number of permutations in which the groom's mother is not next to the bride is 720 - 240 = 480.
c) Consider this in 5 cases. Call the positions (from left to right) positions 1, 2, 3, 4, 5 and 6. Since the bride's mother must be to the left of the groom, the groom could be in position 2, 3, 4, 5 or 6. Each of these cases is worked out individually in the tables below. The first position to fill is the position of the groom, then the positions to the right of the groom (which cannot include the bride's mother, so there are only four choices of people to the right of the groom) and finally the positions to the left of the groom. In the tables, g stands for groom and m stands for bride's mother.
| position 1 | position 2 | position 3 | position 4 | position 5 | position 6 (g) |
| 5 | 4 | 3 | 2 | 1 | 1 |
| position 1 | position 2 | position 3 | position 4 | position 5 (g) | position 6 |
| 4 | 3 | 2 | 1 | 1 | 4 |
| position 1 | position 2 | position 3 | position 4 (g) | position 5 | position 6 |
| 3 | 2 | 1 | 1 | 4 | 3 |
| position 1 | position 2 | position 3 (g) | position 4 | position 5 | position 6 |
| 2 | 1 | 1 | 4 | 3 | 2 |
| position 1 (m) | position 2 (g) | position 3 | position 4 | position 5 | position 6 |
| 1 | 1 | 4 | 3 | 2 | 1 |
Now these cases do not contain any overlap, so we can use multiplication to find the number of possible arrangements for each case and then add the cases together. Hence there are 5! + (4 · 4!) + (3 · 4!) + (2 · 4!) + (1 · 4!) = 120 + 96 + 72 + 48 + 24 = 360 possible photo arrangements in which the bride's mother is to the left of the groom.