Chapter 1 summary Combinatorial Analysis • Fundamental counting principles: 1. (multiplication principle) Say task 1 can be done in n 1 ways, task 2 can be done in n 2 ways, ..., task k can be done in n k ways. Then if we wish to do all the tasks in sequence, i.e. task 1, followed by task 2, . .. , followed by task k (choosing just one of the available ways to do each of the tasks), then this can be done in n 1 · n 2 · . . . · n k ways . 2. (addition principle) Say we have just one task to do. If we have m ways to do it and also n ways to do it (in which there is no overlap between the two sets of ways of doing it), then the one task can be done in m + n ways . • Permutations and combinations: Say we have n diﬀerent objects and we wish to select k of those objects. 1. (permutations) If we then wish to arrange those k objects in a row (so each diﬀerent ordering of the objects in a row counts as a diﬀerent selection) then there are P ( n, k ) = n ( n-1)( n-2) . . . ( n-k + 1)
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