In how many ways can 5 boys and 4 girls be arranged on a bench if
(a) Boys and girls are in a separate group
(b) Mary and Phillip wish to stay together
(C) There is no restriction
(d) John and Daniel wish to stay together
a) the boys could be on the left as a group or on the right as a group
number of ways = 2(5!)(4!) = ...
b) they can only be together when they stand next to each other near the
middly
number of ways = 2(4!)(1)(1)(3!) = ...
c) no restriction.... number of ways = 9!
d) you do it, let me know how you did it.
I would assume that case b) doesn't depend on case a), in which case Mary and Phillip can be considered as one person (with 2 ways to position them relative to each other), so:
2(8!) ways.
To find the number of ways to arrange the boys and girls on a bench, we can use the concept of permutations.
(a) If boys and girls are in separate groups, we can calculate the number of ways independently for each group and then multiply the results.
For the boys' group, there are 5 boys, so the number of ways to arrange them is 5!.
For the girls' group, there are 4 girls, so the number of ways to arrange them is 4!.
Therefore, the total number of ways to arrange the boys and girls in separate groups is 5! * 4!.
(b) If Mary and Phillip wish to stay together, we can consider them as one entity. So, we have a total of 8 entities (group of Mary and Phillip, the remaining boys, and the remaining girls) to arrange.
The number of ways to arrange these 8 entities is 8!.
Additionally, within the group of Mary and Phillip, they can be arranged in 2 ways: Mary sitting to the left of Phillip or Mary sitting to the right of Phillip.
Therefore, the total number of ways to arrange the boys and girls with Mary and Phillip together is 8! * 2.
(c) If there are no restrictions, we can simply calculate the total number of ways to arrange all 9 children on the bench.
The number of ways to arrange 9 entities (5 boys, 4 girls) is 9!.
Therefore, the total number of ways to arrange the boys and girls with no restriction is 9!.
(d) If John and Daniel wish to stay together, we can consider John and Daniel as one entity. So, we have a total of 8 entities (group of John and Daniel, the remaining boys, and the remaining girls) to arrange.
The number of ways to arrange these 8 entities is 8!.
Within the group of John and Daniel, they can be arranged in 2 ways: John sitting to the left of Daniel or John sitting to the right of Daniel.
Therefore, the total number of ways to arrange the boys and girls with John and Daniel together is 8! * 2.
I hope this explanation helps you in understanding how to solve each scenario.