F. How Many Ways can You Organize?

1. F. How Many Ways can You Organize? Math 30: Pre-Calculus PC30.12 Demonstrate understanding of permutations, including the fundamental counting principle. PC30.13 PC30.13 Demonstrate understanding of combinations of elements, including the application to the binomial theorem.

2. Key Terms:

4. 1. Permutations • PC30.12 • Demonstrate understanding of permutations, including the fundamental counting principle.

5. 1. Permutations

6. Counting Methods are used to determine the number of members from a set as well as the outcome of an event. • There are methods such as tables, lists or tree diagrams that allow you to visually see all the outcomes. • Another methods for determining the number of outcomes is called the Fundamental Counting Principle.

7. The fundamental counting principle states that if one task can be performed in a ways and second task can be performed in b ways, then the two tasks can be performed in (a)(b) ways.

8. Example 1

9. In the last example when we arranged the 5 students in the middle we end up with (5)(4)(3)(2)(1) ways. • This can be abbreviated as 5! and is read as 5 factorial. • Therefore, 5!= (5)(4)(3)(2)(1)

11. Seven different objects can arranged 7! Ways • If there are 7 members of a student council how many ways can they select the chair, secretary and treasurer?

13. Example 2

14. With permutations we said order of different objects is important. Well what is some of the objects in the set are identical. • Consider the word WEED and all the possible 4 letter arrangements.

15. If all the letters were different the number of outcomes would be 4!=24 • There are however 2 identical letters. If they were different we would arrange them 2!=2 ways. • So the number of arrangements of the word WEED is

16. For permutations with repeating objects, a set of n objects with “a” of one kind that are identical, “b” of a second kind that are identical, and “c” of a third kind that are identical, and so one, can be arranged in

17. Example 3

18. Example 4

19. To solve some problems you must count the different arrangements in all the cases that together cover all the possibilities. • Calculate the number of arrangements for each case and then add that values for all cases to obtain the total number of arrangements. • Whenever you encounter a situation with constraints or restriction, always address the choices for the restricted positions first.

20. For example, you may need to determine the number of arrangements of 4 girls and 3 boys in a row of 7 seats if the end of the rows must be either both male or both female.

21. Example 5

22. Key Ideas p. 526

23. Practice • Ex. 11.1 (p.524) #1,2-8 odds in each, 9-18 evens, 22 #5-8 odds in each, 9, 8-26 evens

24. 2. Combinations • PC30.13 • PC30.13 Demonstrate understanding of combinations of elements, including the application to the binomial theorem.

25. 2. Combinations

26. A combination is a selection of a group of objects taken from a larger group. • The kinds of objects selected is important but NOT the order in which they are selected. • There are a few ways to find the possible number of combinations

27. On is to use reasoning. Use the fundamental counting principle and divide by the number of ways that the objects can be arranged among themselves. • For example, calculate the number of combinations of 3 digits made from 1-5 without repetition.

28. There are 60 ways to arrange 3 items form 5 • However, 3 digits can be arranged 3! Ways among themselves and in a combination there are considered the same selection. • So

30. So number of ways of choosing 3 digits from five digits is:

31. The number of combinations of “n” items taken “r” at a time is equivilent to teh number of combinations of n items taken n-r at a time; that is nCr=nCn-r • Proof:

32. To solve some problems, count the different combinations in cases that together cover all the possibilities. • Calculate the number of combinations for each case and then add the values for all cases to obtain the total number of combinations.

33. Example 1

34. Example 2

35. Example 3

36. When answering questions it is important to know if you are dealing with a permutation or a combination. • Remember in Permutations the order of the objects is important. • IN the combinations the type of objects is important but NOT the order in which they are selected.

37. Key Ideas: p.533

38. Practice • Ex. 11.2 (p.534) #1-6 odds in each, 7-13, 14-20 evens #4-6 odds in each, 7-13, 14-24 evens

39. 3. The Binomial Theorem • PC30.13 • PC30.13 Demonstrate understanding of combinations of elements, including the application to the binomial theorem.

40. 3. The Binomial Theorem

41. If you expand a power of a binomial expression, such as (x+y)4 you get a series of terms • There are many patterns in the expression of (x+y)4

45. Example 1

47. For coefficients you can use Pascal’s triangle instead of Combinations.

48. Important Properties of the binomial expansion (x+y)n include: • Write binomial expansions in descending order of exponent of the first term in the binomial • The number of objects, “k”, selected in the combination nCk can be taken to match the number of factors of the second variable. That is, it is the same as the exponent on the second variable. • The sum of the exponents in any term of the expansion is “n”.

49. The General Term tk+1 has the form: nCk(x)n-k(y)k

50. Example 2