5 men, 1 woman and a chold are to be seated around a round table with seven seats. Find the no. of ways they may have be arranged if the woman and the child must sit together.
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.