The Basic Rules of Counting

8 7 6 336 math 30530 fall 2012 counting september 13

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Unformatted text preview: − k )! September 13, 2013 5 / 12 Selecting k things from n, WITHOUT REPLACEMENT In how many ways can we pull out k distinct items from among n different, distinct objects? Order matters: (the k items have to be lined up in a row) n(n − 1) . . . (n − (k − 1)) = n! (sometimes n Pk ) (n − k )! Example: 1st, 2nd and 3th in race with 8 runners? 8 × 7 × 6 = 336 Math 30530 (Fall 2012) Counting September 13, 2013 5 / 12 Selecting k things from n, WITHOUT REPLACEMENT In how many ways can we pull out k distinct items from among n different, distinct objects? Order matters: (the k items have to be lined up in a row) n(n − 1) . . . (n − (k − 1)) = n! (sometimes n Pk ) (n − k )! Example: 1st, 2nd and 3th in race with 8 runners? 8 × 7 × 6 = 336 Order doesn’t matter: (the k items are thrown together in a bag) n(n − 1) . . . (n − (k − 1)) = k! Math 30530 (Fall 2012) n k Counting (“n choose k ”, sometimes n Ck ) September 13, 2013 5 / 12 Selecting k things from n, WITHOUT REPLACEMENT In how many ways can we pull out k distinct items from among n different, distinct objects? Order matters: (the k items have to be lined up in a row) n(n − 1) . . . (n − (k − 1)) = n! (sometimes n Pk ) (n − k )! Example: 1st, 2nd and 3th in race with 8 runners? 8 × 7 × 6 = 336 Order doesn’t matter: (the k items are thrown together in a bag) n(n − 1) . . . (n − (k − 1)) = k! n k (“n choose k ”, sometimes n Ck ) Example: Top three in race with eight runners? Math 30530 (Fall 2012) Counting 8 3 September 13, 2013 5 / 12 Selecting k things from n, WITHOUT REPLACEMENT In how many ways can we pull out k distinct items from among n different, distinct objects? Order matters: (the k items have to be lined up in a row) n(n − 1) . . . (n − (k − 1)) = n! (sometimes n Pk ) (n − k )! Example: 1st, 2nd and 3th in race with 8 runners? 8 × 7 × 6 = 336 Order doesn’t matter: (the k items are thrown together in a bag) n(n − 1) . . . (n − (k − 1)) = k! n k (“n choose k ”, sometimes n Ck ) Example: Top three in race with eight runners? 8 3 Basic counting rule 3 — The overcount rule If x is an initial count of some set of objects, and each object you want to count ap...
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