Permutation and Combinations Practice [Arranging the reindeer]

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marcelolandaverde

2018. március 27. 12:37

Hi everyone,

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,

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Okapi

2018. április 4. 20:44
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Hello marcelolandaverde,

In 2) all the counted combinations are with Glopin and Lancer together.

Your explanation is the approach to determine a probability by calculating its counter-probability. It is harder to count all the arrangements where Glopin and Lancer are apart, so you count at first all possible arrangements (1) and then substract the arrangements where Glopin and Lancer are together (2+3), what remains is the number of arrangements where they are apart, and that's what you are looking for. The last step is missing in your calculation.

So in step 1) you counted all possible arrangements. In 2) you count the arrangements, where they are together, assuming, they are a single element which you can choose, for if you take the one, you must take the other reindeer too. If you call this element (GL) (derived from their first letters) you can have following combinations:

(GL)QJ (GL)JQ J(GL)Q JQ(GL) Q(GL)J QJ(GL), which are six combinations, where Glopin and Lancer are next to each other.

But as stated in 3), this is only the half of the truth. For of course is (LG) also possible as a combination. Imagine to every arrangement above another with Glopin and Lancer standing the other way round, so you will come to twelve possible combinations, which are all possible combinations with Glopin and Lancer standing next to each other.

Now you have to make step 4) to calculate how many combinations there are when the both fighting reindeers stand apart, as was the problem to solve. You take all possible cominations (see step 1: 4*3*2*1 = 24) and subtract the number of combinations where they stand together (see step 3: 12).

24-12=12. There are twelve ways to arrange the reindeers so that Glopin and Lancer will not stand side by side.

Hope this helps! Okapi

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Permutation and Combinations Practice [Arranging the reindeer]

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