1. COMP 170 L2 Page 1 L03: Binomial Coefficients l Purpose n Properties of binomial coefficients n Related issues: the Binomial Theorem and labeling

2. COMP 170 L2 Page 2 Outline l Basic properties l Pascal’s triangle l The Binomial theorem l Labeling and Trinomial coefficients

3. COMP 170 L2 Page 3 Basic properties

4. COMP 170 L2 Page 4 Basic Properties l Correct, but not so telling.

5. COMP 170 L2 Page 5 Proof of.

6. COMP 170 L2 Page 6 Proof of.

7. COMP 170 L2 Page 7 Proof of.

8. COMP 170 L2 Page 8 Basic Properties l Example

9. COMP 170 L2 Page 9 Proof of

10. COMP 170 L2 Page 10

11. COMP 170 L2 Page 11 Proof of

12. COMP 170 L2 Page 12 Proof of

13. COMP 170 L2 Page 13 Summary of Basic Properties

14. COMP 170 L2 Page 14 Outline l Basic properties l Pascal’s triangle l The Binomial theorem l Labeling and Trinomial coefficients

15. COMP 170 L2 Page 15 Pascal’s Triangle

16. COMP 170 L2 Page 16 Pascal’s Triangle l Each entry = sum of the two entries above it

17. COMP 170 L2 Page 17 Pascal’s Triangle l Each entry = sum of the two entries above it l Next row?

18. COMP 170 L2 Page 18 Pascal Relationship l Examples

19. COMP 170 L2 Page 19 Algebraic Proof of Pascal’s Relationship l For reference only. l Will give proof by sum principle. More revealing.

20. COMP 170 L2 Page 20 Proof of Pascal’s Relationship by Sum Principle

21. COMP 170 L2 Page 21 Proof of Pascal’s Relationship by Sum Principle

22. COMP 170 L2 Page 22

23. COMP 170 L2 Page 23 Pascal Relationship

24. COMP 170 L2 Page 24 Outline l Basic properties l Pascal’s triangle l The Binomial theorem l Labeling and Trinomial coefficients

25. COMP 170 L2 Page 25 Expanding Binomials

26. COMP 170 L2 Page 26 The Binomial Theorem l We are concerned with n What is the theorem true?

27. COMP 170 L2 Page 27 Examples l Monomial terms: n Lists of length two, each element can either be x or y. l How many monomial terms with one y (and hence one x) ? n = number of ways to choose 1 place among 2 places n That is the coefficient for the term l Similarly n Coefficient for  = number of lists having 0 place for y = n Coefficient for  = number of lists having 2 places for y = l So

28. COMP 170 L2 Page 28 Examples l Coefficient for n = number of ways to choose 2 places for 3 places. l Coefficient for n = number of ways to choose i places from 3 places

29. COMP 170 L2 Page 29 Proof of the Binomial Theorem l Coefficient of n = number of lists having y in k places n =number of ways to choose k places from n places n=n=

30. COMP 170 L2 Page 30 Applications of the Binomial Theorem

31. COMP 170 L2 Page 31 Applications of the Binomial Theorem

32. COMP 170 L2 Page 32 Outline l Basic properties l Pascal’s triangle l The Binomial theorem l Labeling and Trinomial coefficients

33. COMP 170 L2 Page 33 Labeling with 2 Colors

34. COMP 170 L2 Page 34 Labeling with 3 Colors

35. COMP 170 L2 Page 35 Trinomial Coefficients

36. COMP 170 L2 Page 36 Number of Partitions

37. COMP 170 L2 Page 37 Trinomial Coefficients The number of ways to partition a set of n places into 3 subsets of k1, k2 and k3 places Each list is of length n, consisting of x, y, z