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

  • March 30th-April 3rd Warm-ups - North Bend Middle School

    March 30th-April 3rd Warm-ups - North Bend Middle School

    Which is the longest poem: an epic, a ballad or a sonnet? What European country awards Nobel prizes each year? What's the total number of degrees in any triangle? What system distributes oxygen and nutrients to our body cells? If...
  • Chapter 10: Contract Performance, Breach, and Remedies  2013

    Chapter 10: Contract Performance, Breach, and Remedies 2013

    Duress. Forcing a party to enter into a contract under fear or threat makes the contract voidable. ... (iii) when the assignment materially changes rights or duties of obligor, or (iv) when an assignment will significantly change the risk or...
  • HAEI and HAEII course objective #16

    HAEI and HAEII course objective #16

    Define terms common to the discussion of patients with disorders of the endocrine system (especially diabetes mellitus, and abnormalitiesof pituitary, thyroid, adrenal and parathyroid glands) and review basic principles. ... Fad diets, diet therapy for treatment of diabetes and hyperlipidemia...
  • Area Formulas - Weebly

    Area Formulas - Weebly

    Trapezoid (b1 + b2)h. 2. So we need to account for the split base, by calling the top base, base 1, and the bottom base, base 2. By adding them together, we get the original base from the parallelogram.
  • Soil - Georgetown High School

    Soil - Georgetown High School

    Soil texture describes the size of the mineral particles that make up soil. The 3 main grain sizes are sand, silt, and clay. The 3 subcategories of soil are loam, peat, and chalk. Soil Profiles Soils are formed in layers...
  • Google Scholar Citations - Hawler Medical University

    Google Scholar Citations - Hawler Medical University

    scholar.google.com. Sign in with your gmail account. Provide your information (hmu email if available) Check and add your articles. Edit your profile (photo, make your profile public) Verify your "hmu" email. Create alerts. Understand the citation indices. Search Google Scholar
  • What Is Medication Reconciliation (Med Rec)?

    What Is Medication Reconciliation (Med Rec)?

    What Is Medication Reconciliation (Med Rec)? It is the process in which healthcare providers work collaboratively to identify the most comprehensive and accurate list of medications for the patient/resident as they move from one level of care to another.
  • AP Human Geography Models

    AP Human Geography Models

    Geographer: Abdel Omran (not important) Main Ideas: There is a pattern of the causes of death as your move through the DTM. Visual: Real Life Example: Cholera is a big killer is stage 2. World Systems Theory. Geographer: Immanuel Wallerstein.