it is very important to check that we have not overlooked any possible outcome. one visual method of...
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It is very important to check that we have not overlooked any possible outcome. One visual method of checking this is making use of a tree diagram.
Ex. Flip a coin, then roll a dieS = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}
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The Counting Principle: The total number of possibilities for two or more independent events is the product of the number of possibilities for each event. Example: You want to create a computer code using the letters A, B, C, D, E, and F. If letters may be re-used how many possible codes are there to choose from?
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At Johnny’s Burger Place, a customer can get a customized meal by ordering either a turkey burger, chicken burger, hamburger, or garden burger with a side order of potato chips or french fries with a choice of either juice, milk, or soda. •Use a tree diagram to list all the different combinations of a burger, side order and a drink. •Describe ways and give examples of how Johnny could change his menu so that a customer would have 30 different choices.
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Permutations changing the order of elements arranged in a particular order. (ORDER MATTERS!) Example: Using the word BAT, how many three letter combinations can be made? (order matters and no letter may be repeated)
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10P3 = 10 • 9 • 8 = 720
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Factorial (!) the product of a given positive integer multiplied by all lesser positive integers. This is a case of permutations where all of the objects are used.
Example: You want to create a computer code using the letters A, B, C, D, E, and F. This time letters may only be used once. How many possible codes are there to choose from?
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1) If you have a combination lock that contains only the numbers from 0 to 9, and the combination contains three numbers, how many possible combinations exist for this lock (assume numbers can repeat)? 2) There are 7 books on a shelf. How many different ways can you arrange them? 3) How many different ways can we arrange the letters in the word MATH?
10 • 10 • 10 = 103 = 1000
7 • 6 • 5 • 4 • 3 • 2 • 1= 7! = 5040
4 • 3 • 2 • 1= 4! = 24
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n P k =
n!
(n k)!n : total number of objects in a groupk : total number of objects taken from n
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Example: If six divers are entered into a competition, how many possibilities are there for the top three places? (remember order matters!)
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If 40 names are placed in a hat, how many permutations could be made if 15 names are selected? (assume order matters because of the different prized awarded)
22
1540 1026.5!25
!40
!1540
!40
P
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Combinations the arrangement of elements into various groups without regard to their order in the group. Example: Using the word BAT, how many two-letter combinations can be made? (remember order doesn’t matter!)
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n C k =
n!
k!(n k)!
n : total number of objects in a groupk : total number of objects taken from n
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Example: With 32 seeds at Wimbledon (a famous tennis tournament in Europe), how many two player combinations are there for the final match?
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816,983,13123456
444546474849
)!649(!6
!49649
C
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1) How many different ways can you eliminate all of the 16 balls from a pool table (assuming that hitting the 8 ball in early doesn’t end the game like real pool)? Order matters!
2) How many ways can first and second place be awarded to 10 people?
16! = 2.09 • 1013
10P2 = 10 • 9 = 90
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3) Using the word numbers: (a) If order matters, how many different arrangements are there for all letters in numbers? (b) If b was definitely the first letter, now how many possible arrangements are there? 4) You have 5 shirts, but you will select only 3 for your vacation. In how many different combinations of shirts can you bring?
7! = 5040
6! = 720
10123
345
)!35(!3
!535
C