1. Discrete Structures – CNS2300 Text Discrete Mathematics and Its Applications (5 th Edition) Kenneth H. Rosen Chapter 4 Counting

2. Section 4.5 Generalized Permutations and Combinations

3. Permutations with Repitition The number of r -permutations of a set of n objects with repetition is n r Since repetition is allowed, the number of items to select from remains the same for each choice

4. Permutations with Repetition

5. Example A drawing is to be held for three prizes. After a ticket is drawn from a bowl to determine a winner, the ticket is returned to the bowl before another ticket is drawn. In how many ways may the prizes be awarded if there are 100 names in the bowl?

6. Combinations with Repetition Consider a bowl containing a red block and a yellow block. Draw two blocks, with replacement. Since order is not important in combinations, the different combinations are

7. Combinations with Repetition Consider a bowl containing three, different colored cubes. Draw three, with replacement.

8. Combinations with Repetition Eliminate the various permutations.

9. Theorem There are C( n + r - 1, r ) r -combinations from a set with n elements when repetition of elements is allowed.

10. Permutations with Indistinguishable Objects There are six blocks but only two distinguishable colors. The red blocks are indistinguishable

11. Permutations with Indistinguishable Objects One arrangements of the blocks is as follows: Swapping the first and second blocks results in a different, but indistinguishable arrangement.

12. Permutations with Indistinguishable Objects For any arrangement of the blocks, the red blocks could be rearranged in 4! different ways and the blue blocks could be rearranged in 2! different ways and none of them would be distinguishable. If the blocks were all distinguishable then there would be 6! different arrangements. To find the distinguishable arrangements with the given color we must divide

13. Theorem The number of different permutations of n objects, where there n 1 indistinguishable objects of type 1, n 2 indistinguishable objects of type 2,..., and n k indistinguishable objects of type k, is

14. Distributing Objects into Boxes Dealing cards Placing people in different rooms Packing items in different boxes

15. Theorem The number of ways to distribute n distinguishable objects into k distinguishable boxes to that n i objects are placed into box i, i = 1, 2, …, k, equals Look familiar?

16. XYZ 1,2 12 12 21 21 21 12 12 21 21 21 12 12 How many ways are there to assign two jobs fo three employees if each employee can be given more than one job?

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