6.8 - Pascal's Triangle and the Binomial Theorem

6.8 - Pascal's Triangle and the Binomial Theorem

11.1 Pascals Triangle and the Binomial Theorem FACTORIALS, COMBINATIONS Factorial is denoted by the symbol !. The factorial of a number is calculated by multiplying all integers from the number to 1. Formal Definition The symbol n!, is define as the product of all the integers from n to 1. In other words, n! = n(n - 1)(n 2)(n 3) 3 2 1 Also note that by definition, 0! = 1 Example #9

3! 3 2 1 6 (9 3)! 6! 6 5 4 3 2 1 720 9! 9 8 7 6 5 4 3 2 1 362,880 Combinations Definition Combinations give the number of ways x element can be selected from n distinct elements. The total number of combinations is given by, n Cx and is read as the number of combinations of n elements selected x at a time.

The formula for the number of combinations for selecting x from n distinct elements is, Note: n! n Cx x !(n x )! n! n! n! 1

n Cn n !(n n)! n ! 0! n ! n C0 n! n! n! 1 0!( n 0)! 0! n ! n ! Combinations Example #10

2 5! 5! 5 4 3 2 1 10 5 C3 3!(5 3)! 3! 2! 3 2 1 2 1 7! 7! 7 6 5 4 3 2 1

35 7 C4 4!(7 4)! 4! 3! 4 3 2 1 3 2 1 4 C0 1 3 C3 1 The Binomial Theorem Strategy only: how do we expand these?

1. 3. (x + 2)2 (x 3)3 2. 4. (2x + 3)2 (a + b)4 The Binomial Theorem Solutions

1. (x + 2)2 = x2 + 2(2)x + 22 = x2 + 4x + 4 2. (2x + 3)2 = (2x)2 + 2(3)(2x) + 32 = 4x2 + 12x + 9 3. (x 3)3 = (x 3)(x 3)2 = (x 3)(x2 2(3)x + 32) = (x 3)(x2 6x + 9) = x(x2 6x + 9) 3(x2 6x + 9) = x3 6x2 + 9x 3x2 + 18x 27 = x3 9x2 + 27x 27 4. (a + b)4 = (a + b)2(a + b)2 = (a2 + 2ab + b2)(a2 + 2ab + b2) = a2(a2 + 2ab + b2) + 2ab(a2 + 2ab + b2) + b2(a2 + 2ab + b2) = a4 + 2a3b + a2b2 + 2a3b + 4a2b2 + 2ab3 + a2b2 + 2ab3 + b4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 THAT is a LOT of work! Isnt there an easier way?

Introducing: Pascals Triangle Row 5 Row 6 Take a moment to copy the first 6 rows. What patterns do you see?

The Binomial Theorem Use Pascals Triangle to expand (a + b)5. Use the row that has 5 as its second number. The exponents for a begin with 5 and decrease. 1a5b0 + 5a4b1 + 10a3b2 + 10a2b3 + 5a1b4 + 1a0b5 The exponents for b begin with 0 and increase. In its simplest form, the expansion is a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5. Row 5 The Binomial Theorem Use Pascals Triangle to expand (x 3)4.

First write the pattern for raising a binomial to the fourth power. 1 4 6 4 1 Coefficients from Pascals Triangle.

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 Since (x 3)4 = (x + (3))4, substitute x for a and 3 for b. (x + (3))4 = x4 + 4x3(3) + 6x2(3)2 + 4x(3)3 + (3)4 = x4 12x3 + 54x2 108x + 81 The expansion of (x 3)4 is x4 12x3 + 54x2 108x + 81. The Binomial Theorem For any positive integer, n (a b) n a n n! n! n!

n! a n 1b a n 2b 2 a n 3b 3 ... a n r b r ... b n 1!(n 1)! 2!(n 2)! 3!(n 3)! r!(n r )! n! n Cx x !(n x )!

The Binomial Theorem Use the Binomial Theorem to expand (x y)9. Write the pattern for raising a binomial to the ninth power. (a + b)9 = 9C0a9 + 9C1a8b + 9C2a7b2 + 9C3a6b3 + 9C4a5b4 + 9C5a4b5 + 9C6a3b6 + 9C7a2b7 + 9C8ab8 + 9C9b9 Substitute x for a and y for b. Evaluate each combination. (x y)9 = 9C0x9 + 9C1x8(y) + 9C2x7(y)2 + 9C3x6(y)3 + 9C4x5(y)4 + 9C5x4(y)5 + 9C6x3(y)6 + 9C7x2(y)7 + 9C8x(y)8 + 9C9(y)9 = x9 9x8y + 36x7y2 84x6y3 + 126x5y4 126x4y5 + 84x3y6 36x2y7 + 9xy8 y9 The expansion of (x y)9 is x9 9x8y + 36x7y2 84x6y3 + 126x5y4 126x4y5 + 84x3y6 36x2y7 + 9xy8 y9.

Lets Try Some Expand the following a) (x-y5)3 b) (3x-2y)4 Lets Try Some Expand the following (x-y5)3 Lets Try Some Expand the following

(3x-2y)4 Lets Try Some Expand the following (3x-2y)4 How does this relate to probability? You can use the Binomial Theorem to solve probability problems. If an event has a probability of success p and a probability of failure q, each term in the expansion of (p + q)n represents a probability.

Example: 10C2 * p8 q2 represents the probability of 8 successes in 10 tries The Binomial Theorem Brianna makes about 90% of the shots on goal she attempts. Find the probability that Bri makes exactly 7 out of 12 consecutive goals. Since you want 7 successes (and 5 failures), use the term p7q5. This term has the coefficient 12C5. Probability (7 out of 10) = 12C5 p7q5 12! = 5! 7! (0.9)7(0.1)5

The probability p of success = 90%, or 0.9. = 0.0037881114 Simplify. Bri has about a 0.4% chance of making exactly 7 out of 12 consecutive goals.

Recently Viewed Presentations

  • 1 What Prayer Will And Will Not Do

    1 What Prayer Will And Will Not Do

    1. Prayer can strengthen the soul. Ps. 138:1-3 "I will praise thee with my whole heart: before the gods will I sing praise unto thee. I will worship toward thy holy temple, and praise thy name for thy lovingkindness and...
  • Role of Women

    Role of Women

    Role of Women created by: Brendan D. and David H Famous Women of the Time Anna Comnena Heloise Hildegard of Bingen Julian of Norwich Christine de Pizan Jane Shore Alice Perrers Katherine Swynford Margery Kempe Joan of Arc Lady Godiva...
  • Thor and Loki Thor and Loki are brothers,

    Thor and Loki Thor and Loki are brothers,

    Battle 2. F. ifteen more of Loki's recruits have surrounded Thor in a royal outbuilding. His hammer will defeat anyone within 4 metres of the outbuilding.
  • Binding the Corporation - Wake Forest University

    Binding the Corporation - Wake Forest University

    Corporations: A Contemporary Approach Chapter 8 Actions Binding the Corporation Slide * of 23 Paul Klee, Monument (1929) (inverted) * * * * * * * * * * * * * * * * * * * * *...
  • GCxGC workshop - ANALYTICAL METHODOLOGY CENTRE

    GCxGC workshop - ANALYTICAL METHODOLOGY CENTRE

    More recently multidimensional gas chromatography (MDGC) was developed as a technique following the ideas and proposals of several groups and Giddings in 1995 evaluated sample dimensionality as a predictor of order-disorder in component peak distribution in multidimensional separation.
  • Meritocracy - Short Cuts

    Meritocracy - Short Cuts

    Inequality is necessary and universal because all societies have inequalities. If inequality exists, then it must be because it is for the good of society. This justifies the high pay and status of the richest people in society.
  • Design Method of Data Collection Surveys and Polls

    Design Method of Data Collection Surveys and Polls

    Non-respondents tend to behave differently to respondents with respect to the question being asked. 1936 U.S. Election Country struggling to recover from the Great Depression 9 million unemployed 1929-1933 real income dropped by 1/3 1936 U.S. Election Candidates: Albert Landon...
  • Newbery Award Winners - EPG Junior High Scholastic Bowl

    Newbery Award Winners - EPG Junior High Scholastic Bowl

    Newbery Award Winners. 1997 Medal Winner: The View from Saturday by E.L. Konigsburg 1996 Medal Winner: The Midwife's Apprenticeby Karen Cushman 1995 Medal Winner: Walk Two Moonsby Sharon Creech The View-HOW HAD MRS. OLINSKI CHOSEN her sixth-grade Academic Bowl team?