Unit 9 Day 4 Agenda Questions from Counting (last class)? Recall Combinations and Factorial Notation 1. Simplify:!! 2. Simplify: 2 Recall (a + b) n Sec 12.6 un9act4: Binomial Experiment pdf version template Post Binomial Theorem and Pascal's Triangle Lecture pdf version Optional: #6 of 2009 summer math is similar to #5 in the lecture notes which led me to thinking about a Trinomial Pyramid Prepare for Unit 9 Test Finish Unit 9 Skills Finish Unit 9 E-Hmwk Finish other Unit 9 Practice Problem Sheets Work on the Unit 9 Review Guide For Tonight's Homework unit 9 skills 5: Binomial Theorem and Pascal's Triangle D2L hmwk 9, page 4 OR WebAssign sec. 12.6 Recommended textbook problems: sec 12.6 (11, 13, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41)
Math 122 / 126 Binomial Experiment Activity 1. Calculate: (a + b) 0 = (a + b) 1 = (a + b) 2 = (a + b) 3 = (a + b) 4 = 2. Below, use the formula for combinations to write down your answers in the blanks above the combinatoric notation. 3. Compare your results for number 1 with your results for number 2. What do you notice? Discuss this with a classmate and if you are feeling bold, volunteer to fill out the template on the board. 4. Use your results from above to guess what the exapansion of (a + b) 5 looks like without bothering to use the distributative property. (a + b) 5 =
Math 122 / 126 Post Binomial Experiment Discussion The pattern you found is called. Copy the pattern of the coefficients that you discovered onto the template below and let's fill in a couple more lines by adding the 2 numbers directly above the number we want. How did this compare to the combination notation? Recall that during our last class we discussed the number of ways to get a subset of a set. Let's sum each row of Pascal's Triangle. How does this relate to the number of subsets? Why? When we put the coefficients of (a + b) n together with the numbers we get from the combinations we end up with the Binomial Theorem which says: We can use summation notation to abbreviate this as: Example Use the Coefficients of Pascal's Triangle to expand (x y) 4 Just for fun, you can expand the idea to trinomials: http://laurashears.info/othermath/trinomialpyramid/
Math 122 / 126 Sec. 12.6: Binomial Theorem and Pascal's Triangle to expand a binomial. Example 1: Expand (x 1/2 + 2y) 5 can be used to find a particular term in the expansion of a binomial. Example 2: Find the 10th term of (2a b) 15 to find a term that contains a particular power on the variable. Example 3: Find the coefficient of the term that contains x 3 in the expansion of: (x 1/2 + 2y) 9 as an application to counting. Example 4: Suppose our class is planning a field trip. Everyone is allowed to go, but not everyone has to go. a. How many ways can we take a subset of the students in this class on the trip? b. How many ways can I take at least 3 students on the trip? Example 5: Suppose I have a contest with 2 contestants and 5 prizes. The winner of each of 5 rounds gets a prize. How many ways can I distribute the 5 prizes to the two contestants? What does this have to do with the Binomial Theorem?
Math 122/126 Unit 9 Skills 4: Binomial Theorem and Pascal s Triangle Pascal's Triangle Problems 1.) Write the next 3 lines of Pascal's Triangle: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 2.) Show that 3.) Use your answer to number 1 and the Binomial Theorem to determine expand (2-3x) 6. 4.) Evaluate the coefficient of x 6 in the expansion of (2 + 0.5x 2 ) 14 5.) Solve for x in: 6 k = 0 6 5 6 k k 729 k x = 6.) a.) How many ways can at most 5 students from a class of 8 equally qualified students go to represent us in a math contest? b.) How many ways can any number of students from the class go including none and all go?