Binomial Expansion And More

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Presentation transcript:

Binomial Expansion And More Jeffrey Bivin Lake Zurich High School Jeff.bivin@lz95.org Last Updated: May 2, 2011

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

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

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

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

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

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

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

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

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

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

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

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 ?

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 ?

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

(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

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

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 ?

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 ?

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 ?

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.

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

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?

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?

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?