Permutations and combinations

The first instance of  is under the Probability and statistics Math Mission. This exercise practices more advanced counting techniques, occasionally mixing in combinations with permutations.

Types of Problems
There are four types of problems in this exercise:


 * 1) Fighting reindeer: This problem describes reindeer that need to be put in a row but some are enemies and cannot be next to each other. The student figures out how many arrangements are possible and places the answer in the box.Pc1.png
 * 2) Reindeer games: This problem describes reindeer that need to be put in a row but some are friends and must be next to each other. The student figures out how many arrangements are possible and places the answer in the box.Pc1-5.png
 * 3) Rearrange the letters: This problem asks how many ways the letters in a certain word with repeated letters can be rearranged. The student finds the answer and writes it in the box.Pc2.png
 * 4) Multiples: This problem asks how many multiples of certain numbers are below one hundred. The student finds the answer and writes it in the box.Pc3.png

Strategies
This exercise is easy to get accuracy badges once you get a feel for the different types of problems and what you need to do on each. The speed badges are medium because there are several different types of problems so some care needs to be taken.


 * 1) If reindeer are enemies, imagine lining up all the reindeer and then subtracting the arrangements where the feuding reindeer are next to each other.
 * 2) If reindeer are buddies, or need to be next to each other, imagine placing the pair of buddies as a group and thus finding the number of rearrangements of one less than the total number of reindeer. Then multiply by two because the reindeer buddies can be reversed in order.
 * 3) Rearrange the letters in Rearrange the letters like normal, but divide by the factorial of the repeated letters since those can be placed in any order and are thus indistinguishable.
 * 4) The quantity of numbers less than one hundred that are multiple of n is the integer part of $$100/n$$.
 * 5) On Multiples don't forget to subtract out the numbers that are multiples of both that you may have double counted.