Honors Precalculus Chapter 9 Summary Basic Combinatorics


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1 Honors Precalculus Chapter 9 Summary Basic Combinatorics A. Factorial: n! means 0! = Why? B. Counting principle: 1. How many different ways can a license plate be formed a) if 7 letters are used and each letter can be repeated? b) if 7 letters are used and no letter can be repeated? c) if 4 letters followed by 3 digits are used, no letter or digit can be repeated? d) if 7 letters or digits are used, each letter or digit can be repeated, and the first character must be a number? If a procedure P has a sequence of stages S 1, S 2,, S n and if S 1 can occur in r 1 ways S 2 can occur in r 2 ways S n can occur in r n ways Then the number of ways that the procedure P can occur is. C. Permutation: Definition: 1. How many ways can 4 people be lined up in a row for a photograph? 2. How many different words can be form using the word MATHEMATICS? The words don t have to make sense. Honors_Precalculus_Ch. 9_Summary pg 1 of 12
2 3. Sixteen people are trying out for roles as dwarfs in a production of Snow White and the Seven Dwarfs. In how many different ways can the director cast the seven roles? Permutation of an nset: If all the elements of an nset are distinguishable from one another, there are permutations. If an nset contains n 1 objects of a first kind, n 2 object of a second kind, and so on with n 1 + n 2 + n k = n, then the number of distinguishable permutations of the nset is If r objects are taken from a set of n (distinguishable) objects to be arranged in order (0 r n), the number of permutations is D. Combination: Definition: Example: 1. Five cards are selected from a deck of 52 cards to form a poker hand. How many different ways can this be done? 2. In the California Super Lotto Plus, five numbers are chosen from 147, no number is repeated and the order does not matter; also, one MEGA number is chosen from a) How many different tickets can be formed? b) You win a $1 if your ticket matches three of the first five numbers. How many different tickets of this kind are possible? The number of combinations of n objects taken r at a time is Honors_Precalculus_Ch. 9_Summary pg 2 of 12
3 Binomial Theorem A. Pascal s Triangle Row 0: Row 1: Row 2: Row 3: Row 4: Describe the patterns in the Pascal s Triangle: Recursion Formula for Pascal s Triangle: " n% " $ ' = % " $ ' + % $ ' or n C r = # r& # & # & B. Binomial Expansion: (a + b) 0 (a + b) 1 + (a + b) (a + b) (a + b) Describe the patterns in the Binomial Expansion, relate it to the Pascal s Triangle: Honors_Precalculus_Ch. 9_Summary pg 3 of 12
4 The Binomial Theorem: For any positive integer n (a + b) n = The rth term in the expansion of (a + b) n = Practice: 1. Expand the binomial: a) (x y) 7 b) (2x + y ) 5 c) Find the 5 th term in the expansion of (x 1) 6 Honors_Precalculus_Ch. 9_Summary pg 4 of 12
5 Probability A. Probability of an event: Define probability of an event in your own words. Give at least one example. What is the difference between empirical probability and theoretical probability: B. Visualizing Probability: Venn diagram & Tree diagram 1) In a high school, 54% of the students are girls and 62% of the students play sports. Half of the girls at the school play sports. Use a Venn diagram to answer the following questions: a) What percentage of the students who play sports are boys? b) If a student is chosen at random, what is the probability that it is a boy who does not play sports? 2. If it rains tomorrow, the probability is 60% that Mrs. Vu will stay home and watch a movie. If it does not rain tomorrow, there is a 10% chance that she will stay home and watch a movie. Suppose that the chance of rain tomorrow is 80%, what is the probability that Mrs. Vu will stay home and watch a movie? Use a tree diagram or a table. Honors_Precalculus_Ch. 9_Summary pg 5 of 12
6 C. Probability of independent events: 1) What is the probability that a family having four boys in a row? 2) Two dice are rolled. What is the probability of getting an even number on one die and a 1 on the other? Two events are independent if If A & B are two independent events then P(A and B) = p(a IB) = D. Probability of mutually exclusive events: 1) If a card is randomly selected from a standard deck of cards, a) what is the probability of getting a king or a queen? b) what is the probability of getting a king or a heart? 2) A group of people is comprised of 25 U.S. men, 20 U.S. women, 10 Canadian men, and 5 Canadian women. If a person is selected at random from the group, find the probability that the selected person is a man or a Canadian. Two events are mutually exclusive if If A & B are two events then P(A or B) = p(a U B) = If A & B are mutually exclusive then p(a IB) = so P(A or B) = p(a U B) = Honors_Precalculus_Ch. 9_Summary pg 6 of 12
7 E. Conditional Probability: 1) Two identical jars are on a table. Jar A contains 10 chocolate chip cookies and 5 peanut butter cookies. Jar B contains 5 chocolate chip cookies and 20 peanut butter cookies. What is the probability that a cookie selected at random is a chocolate chip cookie? 2) A particular drug test is 3% likely to produce a false positive result, that is, of 100 people that have not used drug were tested, the test reports would show 3 positive cases. The same drug test is 5% likely to produce a false negative result, that is, of 100 people that have used drug were tested, the test reports would show 5 negative cases. At a particular high school, the probability of a student having used drugs is 2%. What is the probability that a student tested positive actually used drugs? If event B is dependent on event A, then the probability of B occurring given that A occurred is p(b A) = Honors_Precalculus_Ch. 9_Summary pg 7 of 12
8 E. Binomial Distribution (Bernoulli principle) 1) A family has five children. a) What is the probability of having all boys? b) What is the probability of having exactly one boy? c) What is the probability of having exactly three boys? 2. Mrs. Vu gives her class a test of 15 multiplechoice questions. Each question has 4 answer choices, with only one correct choice. A student guesses randomly on all the questions. a) What is the probability of getting all 15 questions answered correctly? b) What is the probability of getting exactly 10 questions answered correctly? c) What is the probability of getting at least 10 questions answered correctly? Suppose an experiment consists of n independent repetitions of an experiment with two outcomes, called success and failure. Let p(success) = p and p(failure) = q. The probability of getting r successes out of n independent repetitions is Honors_Precalculus_Ch. 9_Summary pg 8 of 12
9 Sequences and Series A. Arithmetic Sequences: 1) a) Find the 100 th term of the arithmetic sequence: 1, 4, 7, 10, b) Define the value of the nth term of the above sequence explicitly. c) Define the value of the nth term of the above sequence recursively d) Find the sum of the first 100 terms. Arithmetic Sequence: A sequence {a n } is an arithmetic sequence if it can be written in the form Each term in an arithmetic sequence can be obtained by: Recursive formula: Explicit formula: Sum of a finite arithmetic sequence: Honors_Precalculus_Ch. 9_Summary pg 9 of 12
10 B. Geometric Sequences: 1) a) Find the 100 th term of the geometric sequence: 1, 2, 4, 8, 16, b) Define the value of the nth term of the above sequence explicitly. c) Define the value of the nth term of the above sequence recursively d) Find the series/sum of the first 100 terms. e) Find a formula for the sum of the first n terms of a geometric sequence. " e) Find # an for the sequence in part a. n=1 " f) Find # an for the sequence 1 n=1 2, 1 4, 1 8, Honors_Precalculus_Ch. 9_Summary pg 10 of 12
11 Geometric Sequence: A sequence {a n } is an arithmetic sequence if it can be written in the form Each term in an arithmetic sequence can be obtained by: Recursive formula: Explicit formula: Sum of a finite geometric series: Sum of an infinite geometric series for r < 1. C. Fibonacci sequence: Honors_Precalculus_Ch. 9_Summary pg 11 of 12
12 Mathematical Induction A. What is it for? B. Procedure for Mathematical Induction Proof: Let P n be a statement about the integer n. Step 1: Prove that P 1 is true. Step 2: Assume that P k is true, prove that P k+1 is true. 1. Prove that (2n 1) = n 2 is true for all positive integers n. Step 1: Show that the statement is true for n = 1. Step 2: Assume that the statement is true for n = k, show that the statement is true for n = k Prove that 3 is a factor of n 3 + 2n. Honors_Precalculus_Ch. 9_Summary pg 12 of 12
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