Suppose that 7 knights and 4 ladies are to be seated at a round table. Find the number of ways this can be done in each of the following cases: a) There are no restrictions; b) No 2 ladies are adjacent; c) The 4 ladies form a single block
To solve this problem, we will use the concept of permutations.
a) When there are no restrictions, we can simply treat the knights and ladies as indistinguishable objects. Therefore, the total number of ways to seat them around a round table is (7+4-1)! = 10!, since there are 7 knights and 4 ladies.
b) When no 2 ladies can be adjacent, we can consider the ladies as objects and place them first. We need to find the number of ways to arrange the 4 ladies such that no two are next to each other.
Starting with 4 empty slots between the ladies, we can place them in the slots. The number of ways to do this is equivalent to the number of ways to arrange 4 objects in the 4 slots, which is 4!.
Now that the ladies are placed, we can arrange the remaining 7 knights among themselves and between the ladies. We have (7+1)! ways to arrange them. Since the knights are indistinguishable from each other, we divide by 7! to account for the repeated arrangements.
Therefore, the total number of ways to seat them is 4! * (8! / 7!).
c) When the 4 ladies form a single block, we can treat them as a single object. This means we only need to arrange 8 objects around the round table (the 7 knights and the single block of ladies).
Therefore, the number of ways to seat them is 8!.