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Permutations and combinations
Description
Exercise Name: Permutations and combinations
Math Missions: High school statistics and probability Math Mission, Precalculus Math Mission, Mathematics III Math Mission
Types of Problems: 4

The Permutations and combinations exercise appears under the High school statistics and probability Math Mission, Precalculus Math Mission and Mathematics III 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 user figures out how many arrangements are possible and places the answer in the box.
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 user figures out how many arrangements are possible and places the answer in the box.
3. Rearrange the letters: This problem asks how many ways the letters in a certain word with repeated letters can be rearranged. The user finds the answer and writes it in the box.
4. Multiples: This problem asks how many multiples of certain numbers are below one hundred. The user finds the answer and writes it in the box.

## Strategies

This exercise is easy to get accuracy badges once users get a feel for the different types of problems and what the user needs 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 ${\frac{100}{n}}$.

## Real-life Applications

1. Probability, along with decimals, percents, and fractions are used to determine the probability of a basketball player making a shot.