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Unformatted text preview: Permutations A permutation of n things taken r at a time, written P ( n, r ), is an arrangement in a row of r things, taken from a set of n distinct things. Order matters. Example 6: How many permutations are there of 5 things taken 3 at a time? Answer: 5 choices for the first thing, 4 for the second, 3 for the third: 5 × 4 × 3 = 60. • If the 5 things are a, b, c, d, e , some possible permuta- tions are: abc abd abe acb acd ace adb adc ade aeb aec aed . . . In general P ( n, r ) = n ! ( n- r )! = n ( n- 1) ··· ( n- r + 1) 1 Combinations A combination of n things taken r at a time, written C ( n, r ) or n r (“ n choose r ”) is any subset of r things from n things. Order makes no difference. Example 7: How many ways are there of choosing 3 things from 5? Answer: If order mattered, then it would be 5 × 4 × 3. Since order doesn’t matter, abc, acb, bac, bca, cab, cba are all the same. • For way of choosing three elements, there are 3! = 6 ways of ordering them. Therefore, the right answer is (5 × 4 × 3) / 3! = 10: abc abd abe acd ace ade bcd bce bde cde In general C ( n, r ) = n ! ( n- r )! r ! = n ( n- 1) ··· ( n- r + 1) /r ! 2 More Examples Example 8: How many full houses are there in poker? • A full house has 5 cards, 3 of one kind and 2 of an- other. • E.g.: 3 5’s and 2 K’s. Answer: You need to find a systematic way of counting: • Choose the denomination for which you have three of a kind: 13 choices. • Choose the three: C (4 , 3) = 4 choices • Choose the denomination for which you have two of a kind: 12 choices • Choose the two: C (4 , 2) = 6 choices....
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- Spring '07
- Binomial Theorem