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Counting Rules

Counting Rules. Basic Counting Rule.

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Counting Rules

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  1. Counting Rules

  2. Basic Counting Rule If r actions are to be performed in a specific order and if there are m1 possibilities for the first action, m2possibilities for the second action, m3 possibilities for the third action and so on to mr possibilities for the last action then the total number of possible outcomes is m1∙m2∙m3 . . . ∙mr The product of m1 through mr. This is known as the Basic Counting Rule

  3. An Example of the Basic Counting Rule James and Carol are having a house built for them. They can choose from a brick front, a stone front or siding for the front of their house. They can choose windows with shutters or windows without shutters for the windows. Finally, they can choose an oak door with a full window, a mahogany door with a half window or a fiberglass door with no window for the front door. How many different configuration can there be for the front of the house? 3∙2∙3 = 18

  4. Problems Using the Basic Counting Rule Students at Enormous State University are each assigned a student ID consisting of two letters and 3 digits. The first letter can be any one of the 5 vowels A, E, I, O or U and the second letter can be any one of the 21 consonants (assume Y is a consonant). The three digits can each be any of the digits 0 – 9 (repetitions are allowed). How many possible student IDs are there? (click mouse to see answer) If we do not allow digits to be repeated how many possible student IDs are there? (click mouse to se answer)

  5. Factorials Frequently when counting the number of possible outcomes we need to use factorials. A factorial is a product of consecutive integers. k! = k(k-1)(k-2)(k-3) . . .3∙2∙1 5! = 5∙4∙3∙2∙1 = 120 It is also the case thatk! = k(k-1)! = k(k-1)(k-2)!

  6. Finding Factorials with the Calculator The TI-83, TI-83 Plus and TI-84 Plus calculators have a built –in function for calculating factorials . To find it press the MATH key, move the cursor to highlight PRB and select option 4: ! To use the calculator to find 12! start by entering 12 on the home screen of the calculator. Then press MATH, highlight PRB and select 4: ! Doing this will return you to the home screen. The line on the calculator should look like this: 12! Press ENTER and the calculator will return the answer 479001600.

  7. Practice Finding Factorials Use your calculator to find the following factorials. (click mouse to see the answer)

  8. Permutations A permutation of r objects from a collection ofm objects is an ordered arrangement of the r objects. It is essential to remember that with permutations order is important. The notation that we use to represent combinations is mPr As an example of some permutations choose any three letters from the alphabet to form a “word”. If we don’t allow letters to be repeated in any word then there are 26 choices for the first letter, 25 choices for the second letter and 24 choices for the 3rd letter. By the Basic Counting Rule the total number of words we can form is 26∙25∙24 = 15,600. Words with the same letters in different arrangements are considered to be different words. This means that act is different from cat is different from tac, etc.

  9. Finding the Number of Permutations The formula for finding the number of permutations of m objects taken r at a time is given by the following formula: Going back to the example on the previous slide the number of 3 letter words with no letters repeating is the number of permutations of 26 objects (letters) taken 3 at a time

  10. Using the Calculator to Find Permutations The TI-83, TI-83 Plus and TI-84 Plus calculators have a built –in function for calculating the number of permutations . To find it press the MATH key, move the cursor to highlight PRB and select option 2: nPr. To use the calculator to find the number of permutations of 12 objects taken 5 at a time start by entering 12 on the home screen of the calculator. Then press MATH, highlight PRB and select 2: nPr. Doing this will return you to the home screen. Next enter 5. The line on the calculator should look like this: 12 nPr 5 Press ENTER and the calculator will return the answer 95040

  11. Practice Finding the Number of Permutations Use your calculator to find the following numbers of permutations. (click mouse to see the answers)

  12. Problems with Permutations How many different four-digit ID numbers can be formed if we don’t allow a digit to be repeated within an ID number? (click mouse to see answer) How many different ways can 6 books be arranged on a shelf?

  13. Combinations A combination of r objects from a collection of m objects is an unordered arrangement of the r objects. Remember that with combinations order is not important. The notation that we use to represent combinations is mCr As an example of combinations choose any three books from the 20 books listed in a book club’s catalog. The total number of three-book combinations is 1140.

  14. Finding the Number of Combinations The formula for finding the number of permutations of n objects taken r at a time is given by the following formula: Going back to the example on the previous slide the number of 3 book combinations out of 20 books is the number of combinations of 20 objects (books) taken 3 at a time

  15. Using the Calculator to Find Combinations The TI-83, TI-83 Plus and TI-84 Plus calculators have a built –in function for calculating the number of combinations . To find it press the MATH key, move the cursor to highlight PRB and select option 3: nCr. To use the calculator to find the number of combinations of 12 objects taken 5 at a time start by entering 12 on the home screen of the calculator. Then press MATH, highlight PRB and select 3: nCr. Doing this will return you to the home screen. Next enter 5. The line on the calculator should look like this: 12 nCr 5 Press ENTER and the calculator will return the answer 220.

  16. Practice Finding the Number of Combinations Use your calculator to find the following numbers of combinations. (click mouse to see the answers)

  17. Problems with Combinations How many different five-card poker hands can be dealt using an ordinary 52 card deck? (click mouse to see answer) How many different committees of 8 people can be formed from a group of 25?

  18. Applying the Rules to Probability A committee of 5 people is to be chosen from a group of 25. Madge loves being on committees and she really wants to be chosen for this one. If the committee members are selected at random what is the probability that Madge will be on the committee? (click mouse for answer) The total number of ways of choosing 5 people from a group of 25 is 25C5 = 53130. This is the total number of different committees that could be formed. The number of committees in which Madge is a member would correspond to the number of ways of choosing the remaining 4 committee members from a group of 24 or 24C4 = 10626 The probability that Madge is on a randomly selected committee is P(Madge) = 10626/53130 = .2

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