SOLUTION: Six students are lining up for flag ceremony. If three persons insist on standing next to each other, how many ways can the six students arrange themselves?

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Question 1108562: Six students are lining up for flag ceremony. If three persons insist on standing next to each other, how many ways can the six students arrange themselves?
Found 2 solutions by josmiceli, Edwin McCravy:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
The 3 that must be together can arrange
themselves 6 different ways:
a b c
a c b
b a c
b c a
c a b
c b a
P( 3,1 ) = 3! / 1!
P( 3,1 ) = 6
-----------------------
Each one of these 6 possible groupings can
fit into the arrangements of the other 3 students:
( think of the 3 that must be together as one
student that 6 different identities )
+6%2AP%28+4%2C1+%29+
+6%2A%28+4%21%2F1%21+%29+
+6%2A4%2A3%2A2%2A1+=+144+
144 different ways
------------------------
Definitely get a 2nd opinion if needed

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
The trio that must stand together can be arranged in 3! = 6 ways.

Then for each of those 6 ways, we have 4 "things" to arrange.
[The 4 "things" consist of 3 single persons and 1 trio.]

That's 4! - 24 ways.

Answer:  3!∙4! = 6∙24 = 144 

Edwin