1. pencil, red pen, highlighter, GP notebook, calculator
U11D11
Have out:
Bellwork:
Expand and simplify (a + 4b)3. 1
Start with row 3 of Pascal’s triangle. +1 1 1
Write powers of a in descending order. +1 1 2 1 +1
Write powers of 4b in ascending order. 1 3 3 1 1 a3
(4b)0 + 3 a2
(4b)1 + 3 a1
(4b)2 + 1 a0
(4b)3 a3
a2b
ab2 b3 +1 +1 +1 +1
total:

2. Yesterday we used Pascal’s Triangle to find the coefficients of the terms of a binomial expansion.
Example: (a + b)4 1 2 3 4 6
Start with row 4 of Pascal’s triangle.
Write powers of a in descending order.
Write powers of b in ascending order. 1 a4 b0 + 4 a3 b1 + 6 a2 b2 + 4 a1 b3 + 1 a0 b4 a4
+ 4a3b
+ 6a2b2
+ 4ab3
+ b4

3. The coefficients can also be rewritten using combinations.
Use the notation ____________.
(a + b)4 = a4 b0 + a3 b1 + a2 b2 + a1 b3 + a0 b4
Notice some patterns:
The sum of the powers of “a” and “b” equal the binomial’s power.
The “r” and the power of “b” are always equal
This pattern holds true for the expansion of any binomial ______.
(a + b)n

4. Binomial Theorem For every positive integer n, (a + b)n = n n 1 n 2 nn 1 n 2 n
n – 2 n
n – 1 n
Example:
Use the Binomial Theorem to write the expansion of (x + y)5. x5 y0 + x4 y1 + x3 y2 + x2 y3 + x1 y4 + x0 y5 x5
+ 5x4y
+ 10x3y2
+ 10x2y3
+ 5xy4
+ y5

5. Under the expansion, label: 1st term, 2nd term, 3rd term, etc.
+ 5x4y
+ 10x3y2
+ 10x2y3
+ 5xy4
+ y5
1st
term
2nd
term
3rd
term
4th
term
5th
term
6th
term
n + 1
Notice that the number of terms in this expression is ______.
one more
For any binomial, the number of terms is always ___________ than the binomial’s ______.
power
Examples: How many terms are in the expansions for the following binomials?
a) (3x + y)9
b) (2x – y)17
c) (x + 4y)125
10 terms
18 terms
126 terms

6. The Binomial Theorem is also very useful whenever we need to find specific terms in an expansion.
To find the rth term of the binomial expansion (a + b)n, use the formula:
Examples: Find the specified term of each binomial expansion.
a) Fourth term of (x + y)5.
x5 – 4 + 1
y4 – 1 x2 y3
10x2y3

7. To find the rth term of the binomial expansion (a + b)n, use the formula:
Examples: Find the specified term of each binomial expansion.
b) Third term of (x – y)20.
x20 – 3 + 1
(–y)3 – 1
Shortcuts:
Notice that these are always equal.
x18
(–y)2
The powers of x and (-y) must add to 20.
So, the power of x must be 20 – 2 = 18
190x18y2

8. Practice: Find the specified term of each binomial expansion.
a) Fifth term of (x + y)12
b) Second term of (x – y)10
x12 – 5 + 1
y5 – 1
x10 – 2 + 1
(–y)2 – 1 x8 y4 x9
(–y)1
495x8y4
–10x9y
c) Fourth term of (x + 2)8
d) Third term of (2x + 1)6
x8 – 4 + 1
(2)4 – 1
(2x)6 – 3 + 1
(1)3 – 1 x5
(2)3
(2x)4
(1)2
56 • x5 • 8
= 448x5
15 • 16x4 • 1
= 240x4

9. Finish the worksheet
And IC - 125, 126, 131

10.  # of ways to select 4 heads from 8 tosses
1. Use Pascal’s Triangle and/or combinations to determine the following probabilities.
a) P (4 heads in 8 tosses)
 # of ways to select 4 heads from 8 tosses
 total # of possible outcomes in 8 tosses 0H 1H 2H 3H 4H 5H 6H 7H 8H
row 8 

11.  # of ways to select 2 heads from 6 tosses
1. Use Pascal’s Triangle and/or combinations to determine the following probabilities.
b) P (2 heads in 6 tosses)
 # of ways to select 2 heads from 6 tosses
 total # of possible outcomes in 6 tosses 0H 1H 2H 3H 4H 5H 6H
row 6 

12.  # of ways to select 6 heads from 9 tosses
1. Use Pascal’s Triangle and/or combinations to determine the following probabilities.
c) P (6 heads in 9 tosses)
 # of ways to select 6 heads from 9 tosses
 total # of possible outcomes in 9 tosses 0H 1H 2H 3H 4H 5H 6H 7H 8H 9H
row 9 

13. d) P (at most 2 heads in 5 tosses)
1. Use Pascal’s Triangle and/or combinations to determine the following probabilities.
d) P (at most 2 heads in 5 tosses) + + 0H 1H 2H 3H 4H 5H
row 5 

14. e) P (at least 5 heads in 7 tosses)
1. Use Pascal’s Triangle and/or combinations to determine the following probabilities.
e) P (at least 5 heads in 7 tosses) + + 0H 1H 2H 3H 4H 5H 6H 7H
row 7 

15. f) P (odd number of heads in 6 tosses)
1. Use Pascal’s Triangle and/or combinations to determine the following probabilities.
f) P (odd number of heads in 6 tosses) + + 0H 1H 2H 3H 4H 5H 6H
row 6 

16. 2. Determine the probabilities for the following 5–card poker hands using a standard deck of cards.
a) all red
# of possible 5–card hands of that are all red.
P (all red) =
Total # of possible 5–card hands
b) all clubs
# of possible 5–card hands of that are all clubs.
P (all clubs) =
Total # of possible 5–card hands

17. 2. Determine the probabilities for the following 5–card poker hands using a standard deck of cards.
c) 3 diamonds and 2 spades
# of ways to get 3 cards from all diamonds
# of ways to get 2 cards from all spades
P (3♦, 2♠) =
Total # of possible 5–card hands

18. 2. Determine the probabilities for the following 5–card poker hands using a standard deck of cards.
f) 3 two’s, 1 ten, 1 other
Think about the groups you are choosing from.
There are:
four 2’s,
four 10’s,
44 cards that are not 2’s or 10’s
# of ways to get 3 cards from all 2’s
# of ways to choose 1 card from all 10’s
# of ways to choose 1 card from everything else
Total # of possible 5–card hands