Final answer:
To seat 10 persons with two insisting on sitting together, treat the 'couple' as one entity, resulting in 9! ways to arrange them. Since the couple can sit in two different orders, the total number of arrangements is 9! x 2, which equals 725,760 different rows.
Step-by-step explanation:
When calculating the number of different ways to arrange 10 people in a row with the restriction that two specific persons must sit together, we can treat these two persons as a single entity. Therefore, we have now 9 entities to arrange - the 'couple' and the remaining 8 individuals. This can be done in 9! (9 factorial) ways. However, within this 'couple', there are 2 ways they can arrange themselves, so we must account for this by multiplying by 2. Therefore, the total number of arrangements will be 9! × 2.
The calculation for 9 factorial (9!) is as follows: 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 362,880. To incorporate the arrangements of the 'couple', we multiply 362,880 by 2, yielding 725,760 different rows where the two persons will always sit together without any other restrictions.