4.8+Permutations+and+Combinations

math \binom{n}{k} math

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is the notation for choose. a formulaic approach to combinations where the order of the chosen set does not matter.

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math \frac{n!}{(n-k)!} math

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is the formula used for permutations. This formula is the mathematical expression for

n!*(n-1)!*(n-2)!... (n-k+1)!

This is usefull for counting how many things are in a set where the order of the set items matter.

Below is a small table comparing Combination and Permutation math



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These properties can be used in conjunction with more complicated procedures such as finding out how many different combinations there are of: 25 juniors running for 3 offices **and** 32 seniors running for 4 offices. you simply multiply

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math \binom{25}{3} \binom{32}{4} math

Another problem discussed in class was the idea of 20 students arranged into 5 groups of 4. The first step is to find how many possible people can go into the first group (that is 20 choose 4) then find how many studens can be in the second group ( ie. 16 choose 4). This process is continued to 4 choose 4 when we realized that this meathod leaves many repeats because it assumes that group order matters. To remove those we divide all of the terms above by 5! the product looks like this math \frac{\binom{20}{4}\binom{16}{4}\binom{12}{4}\binom{8}{4}\binom{4}{4}}{5!} math

The main thing to remember from this is the table above which i will repeat again here due to its significant importance