I am currently learning combinations. When it comes to the practice there is the following problem to solve:
You need to put your reindeer, Quentin, Jebediah, Lancer, and Gloopin, in a single-file line to pull your sleigh. However, Gloopin and Lancer are fighting, so you have to keep them apart, or they won't fly.
How many ways can you arrange your reindeer?
Below you see the solution given:
1)First perform a normal permutation ignoring that you have to keep Gloopin and Lancer apart; so the first part of the solution = 4*3*2*1. So far so gut.
2)We can count the number of arrangements where Gloopin and Lancer are together by treating them as one double-reindeer. Now we can use the same idea as before to come up with 3⋅2⋅1=63, dot, 2, dot, 1, equals, 6 different arrangements. But that's not quite right.
3)Why? Because you can arrange the double-reindeer with Gloopin in front or with Lancer in front, and those are different arrangements! So the actual number of arrangements with Gloopin and Lancer together is 6⋅2=12
Out of this solution I really do not understand the explanation in "2)"; I mean how do you know
which is the arrangement where Glopin and Lancer are together? In general I will be really
thankful if someone could explain me the whole logic ["2)" and "3)"] here.
Thank you in advance,