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Math - Permutations and Combinations
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What is the counting principle in maths?
When there are 'm' ways to do one thing,
and 'n' ways to do another, then there are m×n ways of doing both.
Notice how you multiply the possibilities.
Website:
https://www.mathsisfun.com/data/basic-counting-principle.html
https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/modal/v/tree-diagram-to-count-outcomes
What are permutations?
A collection of things, where the order is important. Permutations refers to arrangement of things.
Example: You want to visit the homes of three friends Alex ("a"), Betty ("b") and Chandra ("c"), but haven't decided in what order. What choices do you have?
Answer: {a,b,c} {a,c,b} {b,a,c} {b,c,a} {c,a,b} {c,b,a}
When the order does not matter, such as in a fruit salad, it is a Combination.
What are combinations?
A collection of things, in which the order does not matter.
Example: For a fruit salad, how many different combinations of 2 ingredients can you make with apple, banana and cherry?
Answer: {apple, banana}, {apple, cherry} or {banana, cherry}
When the order does matter, such as a secret code, it is a Permutation.
What does the factorial operation do and what is its symbol.
The factorial operation has the symbol ! (exclamation mark) when multiplies each consecutive integer below the previous integer all the way up until 1.
E.g. 5! = 5 x 4 x 3 x 2 x 1
What is the permutation formula.
The permutation formula lets us find out the all the possible arrangements of 'n' things in 'k' spots. It gives us all the collections in all of the possible orders. Note that in permutations order matters.
nPk = n!/(n-k)!
nPk means putting 'n' things in 'k' spots considering permutations.
There is also the notation used in this video which is used more often:
https://www.youtube.com/watch?v=JVbbRCVBVRI
What is the combination formula?
The combination formula lets us find all the possible arrangements of different elements.
E.g. lets suppose we have letters ABCDE and we want to put them in 4 spots.
In the permutation world:
ABCD is not the same as ABDC
But in the combination world:
ABCD is the same as ABDC ---> however:
ABCD is not the same as ABCE.
It is just that the collection in combinations has to have at-least one different letter or 'thing'.
Formula --> nCk = n!/k!(n-k)!
Video to explain:
https://www.khanacademy.org/math/statistics-probability/counting-permutations-and-combinations/modal/v/combination-formula
What is 0!?, Explain this result.
0! = 1
This is because if you look at the formula:
nPk = n!/(n-k)!
What if k = n ---> Well then:
nPn = n!/(n-n)! = n!/0!
In order for this to be true and apply with normal permutations, 0! = 1.
E.g. we want to put 3 things in 3 spots. 3! = 3 x 2 x 1 = 6
Or 3P3 = 3!/(3-3)! = 3!/0! = 3!/1 = 3 x 2 x 1 = 6
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