Question

You need to put your reindeer, Lancer, Prancer, Gloopin, and Bloopin, in a single-file line to pull your sleigh. However, Prancer and Lancer are fighting, so you have to keep them apart, or they won't fly.
How many ways can you arrange your reindeer?

This is for Permutations and Combinations.

Answer 1

Answer:

P43=4!(4–3)!=241=24

Step-by-step explanation:

There are four choices you can make for the lead reindeer. For each possible choice, there are then three remaining you can choose to fly second, making 4×3=12 choices for the lead pair. For each possible choice there are two remaining reindeer to take up the back position, making 12×2=24 choices for the team of three.

This type of problem is called a permutation problem, and we write Pnr for the number of ways of choosing r items from n possibilities when the order of the items matters. In this case we are choosing 3 reindeer from 4 possibilities, and the order they appear in the flying line does matter, so the answer we want is P43. The general formula is Pnr=n!(n−r)!. For the answer we are looking for we therefore have:

P43=4!(4–3)!=241=24

Answer 2

Answer:

12

Step-by-step explanation:

Forget about the reindeer that can't be together for a second, and let's try to figure out how many ways we can arrange the reindeer if we don't have to worry about that.

We can build our line of reindeer one by one: there are 444 slots, and we have 444 different reindeer we can put in the first slot.

Once we fill the first slot, we only have 333 reindeer left, so we only have 333 choices for the second slot. So far, there are 4 \cdot 3 = 124⋅3=124, dot, 3, equals, 12 unique choices we can make.

We can continue in this way for the third reindeer, then the fourth, and so on, until we reach the last slot, where we only have one reindeer left and so we can only make one choice.

Hint #44 / 8

So, the total number of unique choices we could make to get to an arrangement of reindeer is 4\cdot3\cdot2\cdot1 = 24.4⋅3⋅2⋅1=24.4, dot, 3, dot, 2, dot, 1, equals, 24, point Another way of writing this is 4!4!4, !, or 444 factorial. But we haven't thought about the two reindeer who can't be together yet.

Hint #55 / 8

There are 24 different arrangements of reindeer altogether, so we just need to subtract all the arrangements where Prancer and Lancer are together. How many of these are there?

We can count the number of arrangements where Prancer and Lancer are together by treating them as one double-reindeer. Now we can use the same idea as before to come up with  6 different arrangements. But that's not quite right.

Why? Because you can arrange the double-reindeer with Prancer in front or with Lancer in front, and those are different arrangements! So the actual number of arrangements with Prancer and Lancer together is 6 \cdot 2 = 126⋅2=126, dot, 2, equals, 12

So, subtracting the number of arrangements where Prancer and Lancer are together from the total number of arrangements, we get 121212 arrangements of reindeer where they will fly.