EXAMPLE 4 Use Pascal’s triangle School Clubs The 6 members of a Model UN club must choose 2 representatives to attend a state convention. Use Pascal’s.

Slides:

Advertisements

Similar presentations

6.1 Use Combinations and Binomial Theorem 378 What is a combination? How is it different from a permutation? What is the formula for n C r ?

Advertisements

Recursion (summary). Two Categories  Translation of recursive definition  Factorial: fact(1) = 1; fact(N) = N * fact(N-1)  Mystery: m(0) = 0; m(1)

EXAMPLE 1 Find combinations CARDS

It’s a triangle. A triangle of numbers! Pascal did not create it…. The Chinese did. Blaise Pascal discovered all of the unique patterns in it.

What percentage of the numbers in Pascal’s Triangle are even?

EXAMPLE 3 Permutations and Combinations Tell whether the possibilities can be counted using a permutation or combination. Then write an expression for.

Counting principle and permutations

EXAMPLE 1 Find probabilities of events You roll a standard six-sided die. Find the probability of (a) rolling a 5 and (b) rolling an even number. SOLUTION.

EXAMPLE 1 Write an equation of a circle Write the equation of the circle shown. The radius is 3 and the center is at the origin. x 2 + y 2 = r 2 x 2 +

F A T R N T EP N I D A Much of mathematics is based on patterns, so it is important to study patterns in math. We use patterns to understand our world.

Binomial Coefficient.

EXAMPLE 5 Find the sum of a series Find the sum of the series. (3 + k 2 ) = ( ) 1 ( ) + ( ) + ( ) + ( ) 8 k – 4 = 19.

The Binomial Theorem.

EXAMPLE 1 Recognizing and Extending a Pattern Scheduling Events You are a member of a summer movie club at your local movie theater. The club meets every.

EXAMPLE 1 Writing Factors Members of the art club are learning to do calligraphy. Their first project is to make posters to display their new lettering.

“Teach A Level Maths” Vol. 1: AS Core Modules

Chapter 11 Counting Methods © 2008 Pearson Addison-Wesley. All rights reserved.

Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 12.3, Slide 1 12 Counting Just How Many Are There?

March 10, 2015Applied Discrete Mathematics Week 6: Counting 1 Permutations and Combinations How many different sets of 3 people can we pick from a group.

Combinations and Pascal’s Triangle 19.0 Students use combinations and permutations to compute probabilities Students know the binomial theorem and.

Binomial Expansion Honors Advanced Algebra Presentation 2-3.

2.4 Use the Binomial Theorem Test: Friday.

Warm Up 2/1/11 1.What is the probability of drawing three queens in a row without replacement? (Set up only) 2.How many 3 letter permutations can be made.

Copyright © 2007 Pearson Education, Inc. Slide 8-1.

Introduction Binomial Expansion is a way of expanding the same bracket very quickly, whatever power it is raised to It can be used in probability, in.

Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4.

Pascal’s triangle - A triangular arrangement of where each row corresponds to a value of n. VOCAB REVIEW:

5-7: The Binomial Theorem

Section 1.1, Slide 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 12.3, Slide 1 12 Counting Just How Many Are There?

At the tip of Pascal's Triangle is the number 1, which makes up the zeroth row. The first row (1 & 1) contains two 1's, both formed by adding the.

EXAMPLE 3 Find possible side lengths ALGEBRA

EXAMPLE 3 Find possible side lengths ALGEBRA A triangle has one side of length 12 and another of length 8. Describe the possible lengths of the third side.

The Binomial Theorem.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Pascal’s Triangle and the Binomial Theorem (x + y) 0 = 1 (x + y) 1 = 1x + 1y (x + y) 2 = 1x 2 + 2xy + 1y 2 (x + y) 3 = 1x 3 + 3x 2 y + 3xy 2 +1 y 3 (x.

EXAMPLE 3 Write recursive rules for special sequences Write a recursive rule for the sequence. a. 1, 1, 2, 3, 5,...b. 1, 1, 2, 6, 24,... SOLUTION Beginning.

Aim: How do we find probability using Pascal’s triangle? Do Now: If a coin is tossed two times, what is the probability you will get 2 heads?

2/16/ Pascal’s Triangle. Pascal’s Triangle For any given value of n, there are n+1 different calculations of combinations: 0C00C00C00C0 1C01C01C01C0.

Lesson  Shown at the right are the first six rows of Pascal’s triangle.  It contains many different patterns that have been studied for centuries.

Section 6.4. Powers of Binomial Expressions Definition: A binomial expression is the sum of two terms, such as x + y. (More generally, these terms can.

Binomial Coefficients and Identities

SATMathVideos.Net A) only I B) only II C) II and III D) I, II and III If two sides of a triangle have sides of lengths 4 and 7, the third leg of the triangle.

EXAMPLE 1 Write an equation of a circle Write the equation of the circle shown. SOLUTION The radius is 3 and the center is at the origin. x 2 + y 2 = r.

Probability Test Review (What are your chances of passing?)

SECTION 10-4 Using Pascal’s Triangle Slide

The Pigeonhole Principle

Using Combinations You have already learned that order is important for some counting problems. For other counting problems, order is not important. For.

WARMUP Lesson 10.2, For use with pages b

The binomial expansions

Use the Binomial Theorem

Use Combinations and the Binomial Theorem

Day 4: Pascal’s Triangle

Use the Binomial Theorem

The Binomial Theorem.

Use the Binomial Theorem

Use Pascal’s triangle to expand the expression (3 x - 2 y) 3

Patterns, Patterns, and more Patterns!

Perimeter and area of Combined shapes

Evaluate the expression.

Chapter 10 Counting Methods 2012 Pearson Education, Inc.

Permutations and Combinations

ALGEBRA 1 CHAPTER # 2 STUDY GUIDE.

Binomial Theorem; Pascal’s Triangle

Chapter 10 Counting Methods.

Adding Like Terms Guided Notes

ALGEBRA II HONORS/GIFTED - SECTION 5-7 (The Binomial Theorem)

Unit 5 Polynomial Operations

ALGEBRA II HONORS/GIFTED - SECTION 5-7 (The Binomial Theorem)

Section 11.7 The Binomial Theorem

Presentation transcript:

EXAMPLE 4 Use Pascal’s triangle School Clubs The 6 members of a Model UN club must choose 2 representatives to attend a state convention. Use Pascal’s triangle to find the number of combinations of 2 members that can be chosen as representatives. SOLUTION Because you need to find 6 C 2, write the 6 th row of Pascal’s triangle by adding numbers from the previous row.

EXAMPLE 4 Use Pascal’s triangle n = 5 (5th row ) n = 6 (6th row ) C06C0 6C16C16C26C26C36C36C46C46C56C56C66C6 ANSWER The value of 6 C 2 is the third number in the 6 th row of Pascal’s triangle, as shown above. Therefore, 6 C 2 = 15. There are 15 combinations of representatives for the convention.

GUIDED PRACTICE for Example 4 WHAT IF? In Example 4, use Pascal’s triangle to find the number of combinations of 2 members that can be chosen if the Model UN club has 7 members. 6. ANSWER 21 combinations