in how many different ways can 3 boys and 2 girls sit on a bench?
in how many different ways can they sit knowing that the boys have to sit near each other and the girls also
what does "near" mean? There are only 5 kids, after all.
To find the number of different ways the boys and girls can sit on a bench, we need to consider two separate scenarios:
Scenario 1: Boys sitting near each other
In this scenario, we can consider the group of three boys as a single entity. So, we have four entities: "BBB" (boys group), "G" (girl 1), "G" (girl 2). The number of ways these entities can be arranged is given by 4!.
However, within the "BBB" group, the three boys can be arranged among themselves in 3! ways. Therefore, the total number of ways the boys and girls can sit when the boys have to sit near each other is: 4! * 3! = 24 * 6 = 144.
Scenario 2: Girls sitting near each other
In this scenario, we can consider the group of two girls as a single entity. So, we have four entities: "B" (boy 1), "B" (boy 2), "B" (boy 3), "GG" (girls group). The number of ways these entities can be arranged is given by 4!.
Within the "GG" group, the two girls can be arranged among themselves in 2! ways. Therefore, the total number of ways the boys and girls can sit when the girls have to sit near each other is: 4! * 2! = 24 * 2 = 48.
To find the total number of ways they can sit, we need to add the results from both scenarios since they are independent of each other. Therefore, the total number of different ways the 3 boys and 2 girls can sit on a bench is: 144 + 48 = 192.