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Miloslav Ciz 2025-03-25 00:54:31 +01:00
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combinatorics.md
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@ -18,7 +18,7 @@ A combination is computed as *C(n,k) = n! / (k! * (n - k)!)*. E.g. having a draw
Furthermore we can define combinations with repetitions in which we allow ourselves to select the same item from the set more than once (note that the selection order still doesn't matter). I.e. while combinations without repetition give us the number of possible subsets, a combinations WITH repetitions gives us the number of possible [multisubsets](multiset.md) of a given set. Combinations with repetition is computed as *Cr(n,k) = C(n + k - 1,k)*. E.g. having a drawer with 4 shirts and wanting to select 2 WITH the possibility to choose one shirt multiple times gives us *Cr(4,2) = C(5,2) = 5! / (2! * (5 - 2)!) = 10* possibilities (AA, AB, AC, AD, BB, BC, BD, CC, CD, DD).
Furthermore if we take combinations and say that order matters, we get generalized permutations that also take two parameters, *n* and *k*, and there are two kinds: without and with repetitions. I.e. permutations without repetitions tell us in how many ways we can choose *k* items from *n* items when ORDER MATTERS, and is computed as *P(n,k) = n!/(n - k)!* (e.g. *P(4,2) = 4!/(4 - 2)! = 12*, AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC). Permutations with repetitions tell us the same thing but we are allowed to select the same thing multiple times, it is computed as *Pr(n,k) = n^k* (e.g. *P(4,2) = 4^2 = 16*, AA, AB, AC, AD, BA, BB, BC, BD, CA, CB, CC, CD, DA, DB, DC, DD).
Additionally if we take combinations and say that order matters, we get generalized permutations that also take two parameters, *n* and *k*, and there are two kinds: without and with repetitions. I.e. permutations without repetitions tell us in how many ways we can choose *k* items from *n* items when ORDER MATTERS, and is computed as *P(n,k) = n!/(n - k)!* (e.g. *P(4,2) = 4!/(4 - 2)! = 12*, AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC). Permutations with repetitions tell us the same thing but we are allowed to select the same thing multiple times, it is computed as *Pr(n,k) = n^k* (e.g. *P(4,2) = 4^2 = 16*, AA, AB, AC, AD, BA, BB, BC, BD, CA, CB, CC, CD, DA, DB, DC, DD).
To sum up: