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COMBINATORICS

COMBINATORICS. Permutations and Combinations. Permutations. The study of permutations involved order and arrangements A permutation of a set of n objects is an ordered arrangement of all n objects. Example.

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COMBINATORICS

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  1. COMBINATORICS Permutations and Combinations

  2. Permutations • The study of permutations involved order and arrangements • A permutation of a set of n objects is an ordered arrangement of all n objects

  3. Example How many 3 letter code symbols can be formed with the letters A,B,C, without repetition? 3!

  4. Factorial Notation nPn = n(n-1)(n-2)...x 3 x 2 x 1 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 0! = 1

  5. Factorial Notation Calculator: Step 1:Number Step 2:MATH Step 3:PRB Step 4: !

  6. The Number of Permutations of n Objects Taken k at a Time nPk = n(n-1)(n-2) ... (n-k-1) = n!/(n-k)! k - factors 8P4 = 8 x 7 x 6 x 5 = 1680 tells how many factors tells where to start

  7. Calculator Step 1: Number n Step 2: MATH Step 3: PRB Step 4: nPr Step 5: Number k

  8. Example 1 How many permutations are there of the letters of the word UNDERMOST if the letters are taken 4 at a time? 9P4 = 3024

  9. Example 2 How many 5-letter code symbols can be formed with letters A,B,C, and D, if we allow a letter to occur more than once? 45 = 1024, which means that...

  10. Repetitions The number of distinct arrangement of n objects taken k at a time, allowing repetition, is nk.

  11. Permutations For a set of n objects in which n1 are one kind, n2 are another kind... and a kth kind, the number of distinguishable permutations is n! n1! x n2! ... x nk!

  12. Example In how many distinguishable ways can the letters of the word CINCINNATI ba arranged? N=10! = 50, 400 2!3!3!1!1!

  13. Combinations(Example) Find all the combinations of 3 letters taken from the set of 5 letters (A,B,C,D,E). There are 10 combinations of the 5 letters taken 3 at a time (A,B,C) names the same set as (A,C,B)

  14. Subset Set A is a subset of set B, denoted A B, if every element of A is a n element of B.

  15. Combinations and Combination Notation A combination containing k objects is a subset containing k objects The number of combinations of n objects taken k at a time is denoted nCk.

  16. Combinations of n Objects Taken k at a Time The total number of combinations of n objects taken k at a time, denoted nCk is given by nCk = n! k!(n-k)!

  17. Example How many committees can be formed from a group of 5 governors and 7 senators if each committee consists of 3 governors and 4 senators? 5C3 x 7C4 = 350.

  18. Binomial Coefficient Notation ( ) = nCk n k

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