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Permutations and Combinations

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Presentation on theme: "Permutations and Combinations"— Presentation transcript:

1 Permutations and Combinations

2 Objectives: Apply fundamental counting principle Compute permutations
Permutations of Similar things Compute combinations distinguish permutations vs combinations

3 Fundamental Counting Principle
Fundamental Counting Principle can be used to determine the number of possible outcomes when there are two or more characteristics . Fundamental Counting Principle states that if an event has m possible outcomes and another independent event has n possible outcomes, then there are m* n possible outcomes for the two events together.

4 Fundamental Counting Principle
Lets start with a simple example. A student is to roll a die and flip a coin. How many possible outcomes will there be? 1H 2H 3H 4H 5H 6H 1T 2T 3T 4T 5T 6T 12 outcomes 6*2 = 12 outcomes

5 Fundamental Counting Principle
For a college interview, Robert has to choose what to wear from the following: 4 slacks, 3 shirts, 2 shoes and 5 ties. How many possible outfits does he have to choose from? 4*3*2*5 = 120 outfits

6 Permutations Notice, ORDER MATTERS!
A Permutation is an arrangement of items in a particular order. Notice, ORDER MATTERS! To find the number of Permutations of n items, we can use the Fundamental Counting Principle or factorial notation.

7 Permutations The number of ways to arrange the letters ABC:
____ ____ ____ 3 ____ ____ Number of choices for first blank? ___ Number of choices for second blank? Number of choices for third blank? 3*2*1 = ! = 3*2*1 = 6 ABC ACB BAC BCA CAB CBA

8 Permutations To find the number of Permutations of n items chosen r at a time, you can use the formula

9 Permutations Practice:
A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated? Answer Now

10 Permutations Practice:
A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated?

11 Permutations Practice:
From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled? Answer Now

12 Permutations Practice:
From a club of 24 members, a President, Vice President, Secretary, Treasurer and Historian are to be elected. In how many ways can the offices be filled?

13 Permutations of Similar things
The number of permutations of n things of which p things are of one kind, q things are of second kind, r things are of third kind and the remaining of different kinds is given by

14 Example Find the number of permutations of all the letters of the word Book & Mississippi For Book = =12 For Mississippi = =34650

15 Permutations when things are Repeated
If two things are to be arranged out of four things A, B, C, D and if things selected for the first time can also be selected for the second time then the no. of arrangements are as follows: AA AB AC AD BA BB BC BD CA CB CC CD DA DB DC DD When r things are to be arranged out of n things and if the repetition of a thing is allowed then the total no. of permutations is nr = 42=16

16 Combinations ORDER DOES NOT MATTER!
A Combination is an arrangement of items in which order does not matter. ORDER DOES NOT MATTER! Since the order does not matter in combinations, there are fewer combinations than permutations.  The combinations are a "subset" of the permutations.

17 Combinations To find the number of Combinations of n items chosen r at a time, you can use the formula

18 Combinations To find the number of Combinations of n items chosen r at a time, you can use the formula

19 Combinations Practice: To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible? Answer Now

20 Combinations To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible? Practice:

21 Combinations Practice: A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions? Answer Now

22 Combinations A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions? Practice:

23 Combinations Practice: A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards? Answer Now

24 Combinations Practice:
A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards? Center: Forwards: Guards: Thus, the number of ways to select the starting line up is 2*10*6 = 120.

25 Thank U

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