1. Binomial Expansion And More
Jeffrey Bivin
Lake Zurich High School
Last Updated: May 2, 2011

2. Look at the exponents! Let’s look at (x + y)p (x + y)0 = 1
(x + y)1 = 1x + 1y
(x + y)2 = 1x2 + 2xy + 1y2
(x + y)3 = 1x3 + 3x2y + 3xy2 +1y3
(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4
(x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5
(x + y)6 = 1x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + 1y6
(x + y)7 = _x7 + _x6y + _x5y2 + _x4y3 + _x3y4 + _x2y5 + _xy6 + _y7

3. Look at the coefficients! Let’s look at (x + y)p (x + y)0 = 1
(x + y)1 = 1x + 1y
(x + y)2 = 1x2 + 2xy + 1y2
(x + y)3 = 1x3 + 3x2y + 3xy2 +1y3
(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4
(x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5
(x + y)6 = 1x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + 1y6

4. Look at the coefficients! Let’s look at (x + y)p (x + y)0 = 1
(x + y)1 = 1x + 1y
(x + y)2 = 1x2 + 2xy + 1y2
(x + y)3 = 1x3 + 3x2y + 3xy2 +1y3
(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4
(x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5
(x + y)6 = 1x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + 1y6

5. Look at the coefficients! PASCAL'S TRIANGLE Let’s look at (x + y)p

6. Let's Apply Pascal's Triangle Let’s look at (x + y)p (x + y)0 = 1
(x + y)1 = 1x + 1y
(x + y)2 = 1x2 + 2xy + 1y2
(x + y)3 = 1x3 + 3x2y + 3xy2 +1y3
(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4
(x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5
(x + y)6 = 1x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + 1y6
(x + y)7 = 1x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y5 + 7xy6 + 1y7

7. In how many ways can you arrange the letters in the word MATHEMATICAL ?

8. In how many ways can you arrange the letters in the non-word xxxxyyy?
In how many ways can you arrange the letters in the non-word xxyyyyy?
(x + y)7 = 1x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y5 + 7xy6 + y7

9. Let’s look closer
(x + y)7 = 1x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y5 + 7xy6 + y7

10. An alternate look
(x + y)7 = 1x7 + 7x6y + 21x5y2 + 35x4y3 + 35x3y4 + 21x2y5 + 7xy6 + y7

11. (2x - y)4 = 16x4 - 32x3y + 24x2y2 - 8xy3 + y4

12. (3x + 2y)5 = 243x5 + 810x4y + 1080x3y2 + 720x2y3 + 240xy4 + 32y5

13. Given: (x + y)15 What is the coefficient of the term ____ x5y10 ?
In how many ways can you arrange the
letters in the non-word
xxxxxyyyyyyyyyy ?

14. Given: (4x - 3y)10 What is the coefficient of the term ____ x7y3 ?
In how many ways can you arrange the
letters in the non-word
xxxxxxxyyy ?

15. x2 + xy + xz + yx + y2 + yz + zx + zy + z2
Expand: (x + y + z)2
(x + y + z) (x + y + z)
x2 + xy + xz + yx + y2 + yz + zx + zy + z2
x2 + 2xy + 2xz + y2 + 2yz + z2

16. (x2 + 2xy + 2xz + y2 + 2yz + z2)(x + y + z)
We did this in the last example
Expand: (x + y + z)3
(x + y + z)2 (x + y + z)
(x2 + 2xy + 2xz + y2 + 2yz + z2)(x + y + z)
x3 + x2y + x2z + 2x2y + 2xy2 + 2xyz
+ 2x2z + 2xzy + 2xz2 + y2x + y3 + y2z +2yzx + 2y2z + 2yz2 + z2x + z2y + z3
Simplify
x3 + 3x2y + 3x2z + 3xy2 + 6xyz
+ 3xz2 + y3 + 3y2z + 3yz2 + z3

17. Given: (x + y + z)3 x3 + 3x2y + 3x2z + 3xy2 + 6xyz
What is the coefficient of the term ____ xyz ?
In how many ways can you arrange the letters in the non-word xyz ?
What is the coefficient of the term ____ x2z ?
In how many ways can you arrange the letters in the non-word xxz ?
x3 + 3x2y + 3x2z + 3xy2 + 6xyz
+ 3xz2 + y3 + 3y2z + 3yz2 + z3

18. Given: (x + y + z)15 What is the coefficient of the term ____ x2y7z6 ?
In how many ways can you arrange the
letters in the non-word
xxyyyyyyyzzzzzz ?

19. Given: (2x + 3y - z)9
What is the coefficient of the term ____ x3y4z2 ?
In how many ways can you arrange the
letters in the non-word
xxxyyyyzz ?

20. Given: (a + b + c + d)20
What is the coefficient of the term ____ a5b6c7d2 ?
In how many ways can you arrange the
letters in the non-word
aaaaabbbbbbcccccccdd ?

21. Binomial Probability
Can be determined in a binomial experiment that meets the following criteria:
► There are n independent trials.
► Each trial has only two possible outcomes:
■ Success
■ Failure
► The probability of success (s) is the same for each trial and the probability for failure (f) is the same for each trail.

22. Binomial Probability
A die is rolled 5 times. What is the probability of rolling exactly 3 ones?

23. Binomial Probability
A bent coin has a probability of heads of 4/7. If the coin is tossed 10 times, what is the probability of tossing exactly 6 heads?

24. Binomial Probability
A bent coin has a probability of heads of 4/7. If the coin is tossed 10 times, what is the probability of tossing at least 8 heads?

25. Binomial Probability
A bent coin has a probability of heads of 4/7. If the coin is tossed 10 times, what is the probability of tossing at least 3 heads?