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Apr 22­6:22 PM

9.1 Basic Combinatorics Name: ______________________

Objectives: Students will be able to use the multiplication principle of counting, permutations or combinations to count the number of ways that a task can be done.

Isaac is a freshman at Kent State University. He is planning his fall schedule for next year. He has the choice of 3 math courses, 2 science courses and 2 humanities courses. He can only select one course from each area. How many course schedules are possible?

Mar 3­1:55 PM

Multiplication Principle of CountingIf a procedure P has a sequence of stages S1, S2, S3, ..., Sn and if

S1 can occur in r1 waysS2 can occur in r2 ways...Sn can occur in rn ways,

then the number of ways that the procedure P can occur is the product r1r2 ...rn.

So, in the case of the college course scheduling, instead of the tree diagram, we could have:

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Apr 22­6:45 PM

Examples:

1.) In this class, how many girl-boy pairs are possible?

2.) If you toss a coin, then roll a 6-sided die and then spin a 4-colored spinner with equal sections, how many outcomes are possible?

3.) How many Ohio license plates are possible if no letter or number can be repeated?

Apr 22­6:59 PM

Factorial Notation: n! = n (n-1) (n-2) 3 2 1Examples: 1.) 3! = 2.) 6! =In the calculator: MATH - PRB - 4: !

There are n! distinguishable permutations of an n-set containing n distinguishable objects.

Examples:1.) How many ways are there to line up the boys in this class?

2.) There are 5 favorite runners in a race. How many ways can the runners win 1st, 2nd, 3rd, 4th and 5th place?

3.) How many ways can the letters of the word MATH be arranged?

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Apr 22­7:28 PM

Examples: 1.) How many ways can the letters of the word CALCULUS be arranged?

2.) How many ways can the letters of the word MISSISSIPPI be arranged?

If an n-set contains n1 objects of a first kind, n2 objects of a second kind, and so on, with n1 + n2 + ... + nk = n, then the number of distinguishable permutations of the n-set is:

____n! n1! n2! ... nk!

Apr 22­6:59 PM

Example: There are ___ boys in this class. How many ways can we select the positions of Math Guru, Math Genius and Math Wizard, assuming that one person cannot hold more than one position?

This is a permutation of ___ objects taken 3 at a time. We need a new rule.

The number of permutations of n objects, taken r at a time, is defined as nPr = n! .

(n - r)!

In the calculator: MATH - PRB - 2: nPr

Note: With permutations: ORDER MATTERS

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Apr 22­7:18 PM

Example: Suppose Mrs. Meinke wants to know the number of possible girl-girl pairs in this class. Why can't a permutation be used?

The number of combinations of n objects taken r at a time is defined as nCr = n! .

(n - r)!r!

In the calculator: MATH - PRB - 3: nCr

Note: With combinations: ORDER DOES NOT MATTER.

Now, using combinations, let's find the number of possible girl-girl pairs.

Mar 3­2:20 PM

Examples

1.) In the Miss America pageant, 51 contestants must be narrowed down to 10 finalists who will compete on national television. In how many possible ways can the 10 finalists be selected?

2.) Sixteen actors answer a casting call to try out for roles as dwarfs in the production of Snow White and the Seven Dwarfs. In how many different ways can the director cast the seven roles?

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Apr 22­7:38 PM

So far, we've been studying objects that are arranged in a line. When objects are arranged in a circle, some of the arrangements are alike.

Example: Consider the problem of making distinct arrangements of 6 people sitting around a table playing cards. How many seating arrangements are possible?

If n objects are arranged in a circle, then there are n! or (n-1)!permutations of the n objects around the circle. n

Mar 8­8:13 AM

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Mar 3­2:27 PM

Example: How many ways are there to place 12 decorative symbols around the face of a clock?

Assignment: Pages 708-710: #1-41 odd, 54,55 Show all work!

Mar 3­2:32 PM

9.1 Group ICE Name: _____________________

1.) The head of a personnel department interviews 8 people for three identical openings. How many different groups of three can be employed?

2.) How many 5 digit numbers exist? (We'll count numbers like 00001 as 5 digits.)

3.) How many different sequences of heads and tails are there if a coin is tossed 10 times?

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Mar 3­2:36 PM

4.) How many ways are there to assign 10 people the roles of president, vice president and treasurer?

5.) How many ways are there to arrange the letters in the word STRAWBERRIES?

6.) Make up a counting problem that has 12C3 as its answer.