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 n P n = 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 n P k = n(n-1)(n-2)... (n-k-1) = n!/(n-k)! k - factors 8 P 4 = 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: n P r 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? 9 P 4 = 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? 4 5 = 1024, which means that...

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

11. Permutations For a set of n objects in which n 1 are one kind, n 2 are another kind... and a k th kind, the number of distinguishable permutations is n! n 1 ! x n 2 !... x n k !

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 n C k.

16. Combinations of n Objects Taken k at a Time The total number of combinations of n objects taken k at a time, denoted n C k is given by n C k = 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? 5 C 3 x 7 C 4 = 350.

18. Binomial Coefficient Notation ( ) = n C k nknk