FUNDAMENTALS OF ALLGEBRA 2A CHAPTER 11 POWERPOINT
PRESENTATION
PERMUTATIONS, COMBINATIONS, AND PROBABILITY
LEARNING TARGETS
• AFTER YOU COMPLETE THIS CHAPTER, YOU WILL BE ABLE TO:
• USE THE FUNDAMENTAL PRINCIPLE OF COUNTING
• DETERMINE THE NUMBER OF PERMUTATIONS AND COMBINATIONS
• FIND THE PROBABILITY OF EVENTS• USE BINOMIAL THEOREM• IDENTIFY SEQUENCES
CHAPTER VOCABULARY
• TREE DIAGRAM: A CHART WITH BRANCHES THAT SHOW RELATIONSHIPS
• FUNDEMENTAL PRINCIPLE OF COUNTING: COUNTING EVENTS: if one task can be completed (p) different ways and a second task can be completed (q) different ways, then the first task followed by the second can be completed in (pq) different ways.
• MULTIPLICATION PRINCIPLE: THE FORMULA TOTAL NUMBER OF SELECTIONS IN SUCCESSION
TREE DIAGRAM
Fundamental Principle of Counting
MULTIPLICATION PRINCIPLE
PERMUTATIONS
• PERMUATIONS: THE ARRANGEMENT OF OBJECTS IN A SPECIFIC ORDER.
PERMUTATION/COMBINATION FORMULA
Combinations
• Combinations: An arrangement of a group of objects in which order is not important.
FORMULA FOR COMBINATIONS
Sample With Replacement
• Suppose you have a bag of marbles: All together there are 10 marbles: 3 are black, 4 are red, 2 are yellow, and 1 is green.
• If you take a marble out and replace it, you will still have:
• 3/10 it is black, 2/5 it is red, 1/5 it is yellow, and 1/10 it is green. The odds do not change when you replace the item you pulled.
BASIC PROBABILITY
• EVENT: AN INDIVIDUAL OUTCOME OF ANY SPECIFIED COMBINATIONS OF OUTCOMES.
• PROBABILITY FRACTION: THE NUMBER OF FAVORABLE OUTCOMES DIVIDED BY THE TOTAL NUMBER OF POSSIBLE OUTCOMES.
• COMPELMENTARY EVENT: AN OUTCOME THAT CAN BE FAVORABLE OR UNFAVORABLE ACCOMPANYING A PARTICULAR EVENT.
SPINNERS, NUMBER CUBES, MARBLES IN BAGS, ETC.
• PROBABILTY EVENTS CAN BE FOUND ON: SPINNERS, MARBLES IN BAGS, NUMBER CUBES(FORMERLY KNOWS AS DICE), LETTERS IN THE ALPHABET, ETC.
BINOMIAL THEOREM
• A WAY TO WRITE COMBINATIONS:
PASCAL’S TRIANGLE
• IF YOU WRITE COEFFICIENTS SEPARATELY, YOU WILL GET A BEAUTIFUL PATTERN KNOWN AS PASCAL’S TRIANGLE:
ARITHMETIC SEQUENCES: VOCABULARY
• SEQUENCE: AN ORDERED LIST OF NUMBERS.• ARITHMETIC SEQUENCE: A SEQUENCE IN
WHICH SUCCESSIVE TERMS DIFFER BY THE SAME NUMBER, (d), CALLED THE COMMON DIFFERENCE.
• COMMON DIFFERENCE: THE CONSTANT, (d), ADDED TO A TERM IN AN ARITHMETIC SEQUENCE TO GET THE NEXT TERM.
MORE VOCABULARY
• SERIES: THE INDICATED SUM OF THE TERMS OF A SEQUENCE.
• DERIVATION: A SEQUENCE OF STATEMENTS THAT SHOWS A RESULT IS A CONSEQUENCE OF ACCEPTED STATEMEMENTS.
• GEOMETRIC SEQUENCE: A SEQUENCE IN WHICH SUCCESSIVE TERMS DIFFER BY THE SAME RATIO (r), CALLED THE COMMON RATIO.
GEOMETRIC MEAN
• GEOMETRIC MEAN: THE SQUARE ROOT OF THE PRODUCT OF TWO NUMBERS. THE GEOMETRIC MEAN BETWEEN A AND B, IS THE SQUARE ROOT OF AB.
• YOU CAN ALWAYS CREATE A GEOMETRIC SEQUENCE OF THREE TERMS BY FINDING THE GEOMETRIC MEAN BETWEEN TWO NUMBERS.
SIGMA NOTATION
• SIGMA IS A LETTER IN THE GREEK ALPHABET AND IT IS USED AS A SYMBOL FOR SUM.
• A SUBSCRIPT IS USED TO SHOW THE FIRST TERM.
• A SUPERSCRIPT IS USED TO SHOW THE LAST TERM.
Sigma Notation
TERMS
• CONVERGENT SERIES: AN INFINITE SERIES WHOSE PARTIAL SUMS APPROACH A FIXED NUMBER AS (n) INCREASES.
• DIVERGENT SERIES: AN INFINITE SERIES WHOSE PARTIAL SUMS DO NOT APPROACH A FIXED NUMBER AS (n) INCREASES.
I LOVE MATH!Permutations!!!!
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