1. A class has four girls and five boys. How many ways can everyone be arranged in a row, so that all four girls are next to each other?
2. A class photo is being taken of six students, sitting in a row. Three of the students are Jamie, Marshall, and Susan, and Jamie wants to sit between Marshall and Susan. (This also means that Jamie is next to both Marshall and Susan.) How many arrangements are possible?
3. A teacher has 15 students in his class, 8 girls and 7 boys. He wants to seat them in three rows of five, but knows that if any row has two girls or two boys sitting next to each other, they will talk and won't pay attention. How many seating arrangements can he make that avoid this?
For 1. :
I have given you some guidelines on how to solve this problem:
We can first think of all the girls as one giant block of G. There are 4 girls, which means within the G, there are ___ ways to arrange all of the four girls. There are ____ ways to arrange the 5 boys (B) and G:
So therefore, __ *__ = your final answer.
For the second one...
There are two choices:
M,J,S and three other people which gives us 24 choices
S,J,M and three other people which gives us 24 choices
24+24 gives us 48 choices in total.
For the first one...
Since four girls are together, the group is basically one person.
Then there are 6 people.
6! is the same as 720.
Then multiply that by 24 again to get the total number of ways to arrange the boys and girls with the constraints.
There are 17280 ways to arrange the boys and girls.