To solve this problem, we need to consider the woman and the child as a single unit. This unit can be seated in one of two ways: either to the left of the woman or to the right.
First, let's consider the case where the woman and child are seated as a single unit to the left of the woman. In this case, the woman-child unit can be considered as one entity, and the remaining 5 men can be arranged in the remaining 6 seats on the table.
The number of ways to arrange the 5 men in 6 seats can be calculated using the formula for permutations:
P(6, 5) = 6!
where "!" denotes factorial. Calculating this, we get:
P(6, 5) = 6 x 5 x 4 x 3 x 2 x 1 = 720
Now, let's consider the case where the woman-child unit is seated to the right of the woman. Again, the woman-child unit can be considered as one entity, and the remaining 5 men can be arranged in the remaining 6 seats on the table. The number of arrangements in this case is also 720.
Since the woman-child unit can be seated in two different ways, we need to multiply the number of arrangements in each case by 2:
Total number of arrangements = 2 x 720 = 1440
Therefore, there are 1440 different ways in which the 5 men, 1 woman, and 1 child can be seated around the round table, with the woman and the child sitting together.