1. 7B and 7C 1 5 10 10 5 1 This lesson is for Chapter 7 Section B
Section C (Binomial Theorem)

2. Binomial Coefficients
The coefficients of the binomial expansion are called binomial coefficients. The coefficients have symmetry.
(x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 1
The first and last coefficients are 1.
The coefficients of the second and second to last terms are equal to n.
Example: What are the last 2 terms of (x + y)10 ? Since n = 10, the last two terms are 10xy9 + 1y10.
The coefficient of xn–ryr in the expansion of (x + y)n is written or nCr .
This is read as, “n choose r.”
Binomial Coefficients

3. Example: Pascal’s Triangle
Example: Use the fifth row of Pascal’s Triangle to generate the sixth row and find the binomial coefficients , , 6C4 and 6C2 .
5th row
6th row 1 6 15 20 15 6 1
6th Row 0th term
6th Row 1th term
6th Row 2th term
6th Row 5th term
6th Row 6th term
6C C C C C4 6C C6
= 6 = and 6C4 = 15 = 6C2.
There is symmetry between binomial coefficients.
nCr = nCn–r
Example: Pascal’s Triangle

4. Example: Pascal’s Triangle
Use the seventh row of Pascal’s Triangle to find the binomial coefficients.
1 1
1th row
2th row
3th row
4th row
0th row 1
8C0 8C1 8C2 8C3 8C4 8C5 8C6 8C7 8C8
Solution:
Example: Pascal’s Triangle

5. Example: Pascal’s Triangle
Find the entire 23rd row of Pascal’s Triangle and circle the 5th coefficient
23C0 23C1 23C2 23C3 23C4 23C5 ….. 23C23
Get your calculators out (Preferably Ti-84)
Type: “23”
Example: Pascal’s Triangle

6. Example: Pascal’s Triangle
23C0 23C1 23C2 23C3 23C4 23C5 ….. 23C23
Example: Pascal’s Triangle

7. Example: Pascal’s Triangle
23C0 23C1 23C2 23C3 23C4 23C5 ….. 23C23 1
Example: Pascal’s Triangle

8. Example: Pascal’s Triangle
23C0 23C1 23C2 23C3 23C4 23C5 ….. 23C23 1 23
Example: Pascal’s Triangle

9. Example: Pascal’s Triangle
23C0 23C1 23C2 23C3 23C4 23C5 ….. 23C23 1 23
253
Example: Pascal’s Triangle

10. Example: Pascal’s Triangle
23C0 23C C C C C5 ….. 23C23
253
1771 1 23
Example: Pascal’s Triangle

11. Example: Pascal’s Triangle
23C0 23C C C C C5 ….. 23C23 1 23
253
1771
8855
Example: Pascal’s Triangle

12. Example: Pascal’s Triangle
23C0 23C C C C C5 ….. 23C23 1 23
253
1771
8855
33649
Example: Pascal’s Triangle

13. Example: Pascal’s Triangle
23C0 23C C C C C5 ….. 23C23 23
253 1
1771
8855
33649 1
Example: Pascal’s Triangle

14. Formula for the Binomial Coefficients
Take a look at Pascal’s Triangle.
Formula for the Binomial Coefficients

15. Let’s hope this works (Please click the picture)

16. Formula for the Binomial Coefficients
How many ways to pick 10 items from 10 items?
Formula for the Binomial Coefficients

17. Formula for the Binomial Coefficients
The symbol n! (n factorial) denotes the product of the first n positive integers. 0! is defined to be 1.
1! = 1
4! = 4 • 3 • 2 • 1 = 24
6! = 6 • 5 • 4 • 3 • 2 • 1 = 720
n! = n(n – 1)(n – 2)  3 • 2 • 1
Formula for Binomial Coefficients For all nonnegative integers n and r,
Example:
Formula for the Binomial Coefficients

18. Example: Binomial coefficients
Example: Use the formula to calculate the binomial coefficients 10C5, 15C0, and
Example: Binomial coefficients

19. Formula for the Binomial Coefficients
How many ways to pick 5 items from 10 items?
Formula for the Binomial Coefficients

20. Definition: Binomial Theorem
Example: Use the Binomial Theorem to expand (x4 + 2)3.
Definition: Binomial Theorem

21. Definition: Binomial Theorem
Although the Binomial Theorem is stated for a binomial which is a sum of terms, it can also be used to expand a difference of terms.
Simply rewrite
(x + y) n as (x + (– y)) n
and apply the theorem to this sum.
Example: Use the Binomial Theorem to expand (3x – 4)4.
Definition: Binomial Theorem

22. Example:Using the Binomial Theorem
Example: Use the Binomial Theorem to write the first three terms in the expansion of (2a + b)12 .
Example:Using the Binomial Theorem

23. Example: Find the nth term
Example: Find the eighth term in the expansion of (x + y)13 .
Think of the first term of the expansion as x13y 0 . The power of y is 1 less than the number of the term in the expansion.
The eighth term is 13C7 x 6 y7.
Therefore, the eighth term of (x + y)13 is 1716 x 6 y7.
Example: Find the nth term

24. Example: Pascal’s Triangle

25. Example: Pascal’s Triangle

26. Let’s try a real IB Paper 1 Question
M10/5/MATME/SP1/ENG/TZ1/XX
Wednesday 5 May 2010 (afternoon)
You are not permitted access to any calculator for this paper.

27. Example: Pascal’s Triangle
Expand (2 + x)4 and simplify your result.
1 point
2 points
Example: Pascal’s Triangle

28. Let’s try another IB Question. Paper 2 (Calculators)
M12/5/MATME/SP2/ENG/TZ1/XX
Friday May 4, 2012 (Morning)
You are permitted access to your calculator for this paper.

29. Example: Pascal’s Triangle
(a) Find b [3 marks]
,.l,
,.l,
,.l,
Example: Pascal’s Triangle

30. Homework Page
17B (1 – 5)
17C (1, 3, 4, 7, 9)

31. Example: Pascal’s Triangle
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Example: Pascal’s Triangle