The Binomial Theorem

1. The Binomial Theorem Macon State College Gaston Brouwer, Ph.D. February 2010

2. Binomial Theorem • Combinatorics • Pascal’s Triangle • Binomial Theorem • Polynomials • Applications • Approximations • Volume of a cube • Questions

3. Combinatorics: Example 1 How many different “words” can you form with the letters: ABC? Solution: 1. ABC 2. ACB 6 different words 3. BAC 4. BCA 5. CAB 6. CBA There are 3 ways to choose the first letter, 2 ways to choose the second letter and 1 way to choose the third letter:

4. Combinatorics: Example 2 How many different “words” can you form with the letters: ABCD? Solution: There are 4 ways to choose the first letter, 3 ways to choose the second letter, 2 ways to choose the third letter, and 1 way to choose the last letter: 24 different words

5. Combinatorics: Example 3 How many different “words” can you form with the letters: AABC? Solution: Unfortunately, we cannot use the reasoning from Example 2. Let’s write out all the possibilities: 1. AABC 2. AACB 3. ABAC 4. ABCA 5. ACAB 6. ACBA 7. BAAC 8. BACA 9. BCAA 10. CAAB 11. CABA 12. CBAA 12 different words

6. Combinatorics: Example 3 (Cont’d) How many different “words” can you form with the letters: AABC? Each of the previous words would lead to 2! = 2 different words if the two A’s would be different letters: 1’a. ADBC 1. AABC 1’. ADBC 1’b. DABC So instead of 4! = 24 words we have: 4 letters total words 2 letters the same

7. Combinatorics “words” of n letters k different letters Theorem 1: The number of distinguishable permutations of objects of different types, where are alike, are alike, …, are alike and is: n = total number of letters number of first letter

8. Combinatorics: Example 4 How many different “words” can you form with the letters: AAABBCC? Solution: Just kidding!  Let’s write out all possibilities: 7 letters total words 3 letters A 2 letters C 2 letters B

9. Combinatorics: Example 5 How many ways can you choose a committee consisting of 3 members from a group of 8 people? Solution: Let’s write this as a “word” problem: The “word” will have 8 letters, representing the 8 people. Let the letter C represent that a person is on the committee and let the letter N represent that a person is not on the committee.

10. Combinatorics: Example 5 (Cont’d) How many ways can you choose a committee consisting of 3 members from a group of 8 people? The word CCCNNNNN represents that persons 1-3 are on the committee and persons 4-8 are not. How many different words are there? 8 letters total words 3 letters C 5 letters N 56 ways

11. Combinatorics from the previous example The fraction calculates how many ways we can choose a group of 3 out of a group of 8 if order is not important. A shorthand notation is: or (read: “8 choose 3”)

12. Combinatorics More general: calculates how many ways we can choose a group of k out of a group of n if order is not important.

13. Combinatorics Some observations: Also: Similarly:

14. Pascal’s Triangle Something really interesting happens when we arrange the expressions in the following way: 0th row 1st row 2nd row 3rd row

15. Pascal’s Triangle Find the 4th and 5th row of Pascal’s Triangle Solution: 4th row

16. Pascal’s Triangle old row Why does this work? new row Claim: Proof:

17. Binomial Theorem

18. Binomial Theorem: Example 6 Expand Solution:

19. Binomial Theorem Why does this work? 3 letters total 3 letters A 0 letters B

20. Binomial Theorem In general: How many different ways can we form the product: “word” m+k total letters Let: Then: is: So the coefficient of m letters A k letters B

21. Binomial Theorem: Example 7 Find the coefficient of the term in Solution: By comparing exponents we see that:

22. Polynomials The binomial theorem can be used to expand polynomials. If is a constant and is a nonnegative integer then is a polynomial in .

23. Polynomials: Example 8 Expand Solution:

24. Polynomials: Example 9 Expand Solution:

25. Aproximations According to the Binomial Theorem we have: If is very small , then is going to be even smaller. So:

26. Aproximations: Example 10 Approximate Solution: Precise answer:

27. Volume of a cube Suppose the dimensions of a cube are by by . What will be the increase in volume if each of the sides of the cube is enlarged by ? Solution: Increase in volume = (new volume) - (old volume)

28. Questions?