Counting

 

   A speech class has $4$ freshmen and $3$ sophomores. (Everyone is distinguishable.)     

   In how many ways can they stand in line, so that all of the freshmen are standing next to each other?     

 

   Mentally strap the freshmen together, temporarily.     

 

   Now you have 4 "things" to arrange (3 sophs and 1 group).     

                  4P4  =  24 arrangements with the freshmen grouped     

 

   But wait.  You can group the freshmen     

                  4P4  =  24 ways to arrange only the group of freshmen     

 

   So, for each arrangement of the entire group, there    

   can be 24 different arrangements of the freshmen.     ==>>     24 • 24 = 576 ways to stand in line     

   _.