1. Red pen, highlighter, GP notebook, calculator
U11D10
Have out:
Bellwork:
Answer the following.
1) P(3 H in 5 tosses) =
2) P(2 H in 9 tosses) = +1 +1 +1 +1 +1 +1
3) P(8 H in 10 tosses) =
4) P(at least 10 H in 12 tosses) = +3 +1 +1 +1 +1 +1
total:

2. Using Pascal’s Triangle for Binomial Expansion
Build a Pascal’s Triangle up to row six. FYI: The “1” at the top is called the ___ row.
row 0 1
row 1 1 1
row 2 1 2 1
row 3 1 3 3 1
row 4 1 4 6 4 1
row 5 1 5 10 10 5 1
row 6 1 6 15 20 15 6 1
Yesterday we discovered that the entries in the triangle represent different ____________________,
Combinations

3. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) nCr
Let’s use another notation for _______________.
combination
nCr =
Rewrite Pascal’s Triangle
using notation up to row 3.
( )
( )
( ) 1 1
( )
( )
( ) 2 2 1 2
( )
( )
( )
( ) 3 3 1 3 2 3

4. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) symmetry
Due to the ___________ of the triangle, we can flip it around.
( )
( )
( ) 1 1
( )
( )
( ) 2 2 1 2
( )
( )
( )
( ) 3 3 2 3 1 3
This is just another way of writing the combination notation. We will apply this notation on day 11.

5. ...and now for something completely different...
Binomial Expansion
Example #1: The binomial we will expand first is (x + 1) 1
= ____
This is the _____ of the Pascal’s Triangle, called row _____.
Top
= _____ + _____
The ___________ of the “expanded” binomial are row ___ of Pascal’s . 1x 1
coefficients 1
1(x)2
2(x) 1
The ___________ of the expanded binomial are row ___ of Pascal’s .
coefficients
= _____ + _____ + _____
x + 1 2 x2 x 1 x +1

6. =
= ( )
x2 + 2x + 1
x2 + 2x + 1
= _____ + _____ + _____ + _____ x x3
2x2 x x2 2x 1
1(x3)
3(x)2
3(x) 1 +1
coefficients 3
The ____________ of the expanded binomial are row ___ of Pascal’s .

7. NOTE: The powers of x match ____ in for each entry. r
Without expanding the binomial the long way, use Pascal’s  to expand the following.
NOTE: The powers of x match ____ in for each entry. r 1
(x + 1)4 = ______ + ______ + ______ + ______ + ______ x4 4 x3 6 x2 4 x1 1 x0
Step 1: list the coefficients of row 4
Similarly, expand:
Step 2: Write the powers in descending order, starting with 4
(x + 1)5 = _____ + _____ + _____ + _____ + _____ + _____ 1 x5 5 x4 10 x3 10 x2 5 x1 1 x0
(x + 1)6 = _____ + _____ + _____ + _____ + _____ + _____ + _____ 1 x6 6 x5 15 x4 20 x3 15 x2 6 x1 1 x0
Make sure to start writing the numbers on the left side of the blanks so that you will have plenty of room!

8. Example #2: Now let’s expand (x + 3) (x + 3)0 = ____
(x + 3)0 = ____
This is row ___ of the Pascal’s . 1 x1 30 x0
(x + 3)1 = _____ + _____
This is row _____ of the Pascal’s . 1 1 31 1 = x
+ 3
(x + 3)2 = _____ + _____ + _____ 1 x2 30 2 x1 31 1 x0 32 2
This is row ____ of the Pascal’s . = x2
+ 6x +9
x + 3
Step 1: list the coefficients of row 2 x
+ 3 x2 3x 9
Step 2: Write the powers of x in descending order, starting with 2.
Step 3: Write the powers of 3 in ascending order, starting with 0.

9. 1 3 3 1
= ______ + ______ + ______ + ______
1. coefficients row 3 1 1 1 1 2 1
Row 3 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

10. = ______ + ______ + ______ + ______1 x3 30 3 x2 31 3 x1 32 1 x0 33
= ______ + ______ + ______ + ______ = x3
+ 9x2
+ 27x
+ 27
1. coefficients row 3
2. powers start with 3
3. powers start with 0
= _______ + _______ + _______ + _______ + _______

11. 1 1. coefficients row 4 1 1 2. powers start with 4 1 2 13 3 1 1 4 6 4 1
Row 4 1 5 10 10 5 1 1 6 15 20 15 6 1 1 x4 30 4 x3 31 6 x2 32 4 x1 33 1 x0 34
= _______ + _______ + _______ + _______ + _______ = x4
+ 12x3
+ 54x2
+ 108x
+ 81

12. Conclusion:
As we move from left to right, the powers of x __________, while the powers of 3 ___________.
decrease
increase
The ____ of the exponents of any term of (x + 3)n is ____.
sum n

13. = _______ + ________ + _______ x2 (-2)0 x1 (-2)1 x0 (-2)2
= ______ + ______ 1 3 3 1 = x
- 2 1 4 6 4 1 = -2
= _______ + ________ + _______ 1 x2
(-2)0 2 x1
(-2)1 1 x0
(-2)2 = x2
– 4x
+ 4

14. = _______ + _______ + _______ + _______ 1 x3 (-2)0 3 x2 (-2)1 3 x1
(-2)2 1 x0
(-2)3 = x3
– 6x2
+ 12x
– 8 -2 = 1 x4
(-2)0 4 x3
(-2)1 6 x2
(-2)2 4 x1
(-2)3 1 x0
(-2)4
= ________ + ________ + ________ + _______ + _______ = x4
– 8x3
+ 24x2
– 32x
+ 16

15. =__________ + _____________
= ___ 1 1
(2x)1
(3)0 1
(2x)0
(3)1
=__________ + _____________ = 2x
+ 3
= _______ + ________ + _______ 1
(2x)2
(3)0 2
(2x)1
(3)1 1
(2x)0
(3)2 =
4x2
+ 12x
+ 9

16. Remember to apply the exponent to the entire term, FOOL!
= _________ + __________ + __________ + _________ 1
(2x)3
(3)0 3
(2x)2
(3)1 3
(2x)1
(3)2 1
(2x)0
(3)3 =
8x3
+ 36x2
+ 54x
+ 27 1
(2x)4
(3)0 4
(2x)3
(3)1 6
(2x)2
(3)2 4
(2x)1
(3)3 1
(2x)0
(3)4
= ________ + ________ + ________ + ________ + _______ =
16x4
+96x3
+216x2
+216x
+81
Remember to apply the exponent to the entire term, FOOL!

17. Example #5: Expand using Pascal’s .
You will run out of room unless you start on the far left of each blank! 1
(3x)4
(-2)0 4
(3x)3
(-2)1 6
(3x)2
(-2)2 4
(3x)1
(-2)3 1
(3x)0
(-2)4
= ___________ + __________ + __________ + _________ + _________ =
81x4
- 216x3
+ 216x2
- 96x
+ 16

18. ROW 3: 1,3,3,1 1 (3x)3 (5)0 + 3 (3x)2 (5)1 + 3 (3x)1 (5)2 + 1 (3x)0
Practice: Completely expand and simplify each binomial using Pascal’s . 1)
ROW 3: 1,3,3,1 1
(3x)3
(5)0 + 3
(3x)2
(5)1 + 3
(3x)1
(5)2 + 1
(3x)0
(5)3
27x x x + 125 2)
ROW 5: 1,5,10,10,5,1 1
(2x)5
(-1)0 + 5
(2x)4
(-1)1 + 10
(2x)3
(-1)2 + 10
(2x)2
(-1)3 + 5
(2x)1
(-1)4 + 1
(2x)0
(-1)5
32x5 – 80x4 + 80x3 – 40x2 + 10x – 1 3)
ROW 4: 1,4,6,4,1 1
(x)4
(-4)0 + 4
(x)3
(-4)1 + 6
(x)2
(-4)2 + 4
(x)1
(-4)3 + 1
(x)0
(-4)4
x4 – 16x3 + 96x2 – 256x + 256

19. 4) ROW 3: 1,3,3,1 1 (3x)3 (-4)0 + 3 (3x)2 (-4)1 + 3 (3x)1 (-4)2 + 1
(-4)3
27x3 – 108x x – 64 5)
ROW 5: 1,5,10,10,5,1 1
(2x)5
(-3y)0 + 5
(2x)4
(-3y)1 + 10
(2x)3
(-3y)2 + 10
(2x)2
(-3y)3 + 5
(2x)1
(-3y)4 + 1
(2x)0
(-3y)5
32x5 – 240x4y + 720x3y2 – 1080x2y xy4 – 243y5

20. Complete the Worksheet.
Don’t be this guy.