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Counting
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Using permutations & combinations alongside principles to count the ways things can be selected or arranged.
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Counting principles
AHL AA 1.10
If we have n ways to do one thing and k to do another, then there are n×k ways to do BOTH (first thing AND the second thing), and n+k ways to do one thing OR the other thing.
Arrangements WITH repetition are k^n
AHL AA 1.10
In any scenario where we can put k items into each of n positions, we have kn possible arrangements.
Ordering items
AHL AA 1.10
The number of different orders in which n items can be arranged is
n1×n−1×⋯2×1=n!🚫
where each box is a "slot" in the order. In the first slot we can put any of n items, in the second any of the n−1 remaining items, and so on. We then multiply all of these together to get n!
Permutations
AHL AA 1.10
The number of permutations, defined as an ordered arrangement, of r items from a set of n items can be calculated by:
nPr=(n−r)!n!📖
Combinations
AHL AA 1.10
Combinations are arrangements of items where order does not matter. For combinations of r items from a set of n:
nCr=r!(n−r)!n!
which is the binomial coefficient.
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