MATH 105: Finite Mathematics 6-5: Combinations Prof. Jonathan - PowerPoint PPT Presentation

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion MATH 105: Finite Mathematics 6-5: Combinations Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Outline Developing Combinations 1 Examples of Combinations 2 Combinations vs. Permutations 3 Conclusion 4

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Outline Developing Combinations 1 Examples of Combinations 2 Combinations vs. Permutations 3 Conclusion 4

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Undoing Order In the last section we found the number of ways to arrange the letters in the word “ninny” as follows. Example Find the number of was to arrange the letters in the word “ninny”.

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Undoing Order In the last section we found the number of ways to arrange the letters in the word “ninny” as follows. Example Find the number of was to arrange the letters in the word “ninny”. P (5 , 5) P (3 , 3)

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Undoing Order In the last section we found the number of ways to arrange the letters in the word “ninny” as follows. Example Find the number of was to arrange the letters in the word “ninny”. P (5 , 5) ← arrange all 5 letters P (3 , 3)

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Undoing Order In the last section we found the number of ways to arrange the letters in the word “ninny” as follows. Example Find the number of was to arrange the letters in the word “ninny”. P (5 , 5) ← arrange all 5 letters P (3 , 3) ← divide out arrangement of 3 n’s

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Undoing Order In the last section we found the number of ways to arrange the letters in the word “ninny” as follows. Example Find the number of was to arrange the letters in the word “ninny”. P (5 , 5) ← arrange all 5 letters P (3 , 3) ← divide out arrangement of 3 n’s Dividing out the order of the n’s is something we can generalize to undoing the order of selection all together.

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Generalizing the Concept Example Suppose that you want to give two movie tickets to your two closest friends. How many ways can you do this? Combinations A combination of n things taken r at a time is the number of ways to select r things from n distinct things without replacement when the order of selection does not matter.

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Generalizing the Concept Example Suppose that you want to give two movie tickets to your two closest friends. How many ways can you do this? P (4 , 2) P (2 , 2) Combinations A combination of n things taken r at a time is the number of ways to select r things from n distinct things without replacement when the order of selection does not matter. P ( n , r ) P ( r , r )

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Generalizing the Concept Example Suppose that you want to give two movie tickets to your two closest friends. How many ways can you do this? P (4 , 2) ← arrange 2 out of 4 people P (2 , 2) Combinations A combination of n things taken r at a time is the number of ways to select r things from n distinct things without replacement when the order of selection does not matter. P ( n , r ) ← arrange r out of n items P ( r , r )

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Generalizing the Concept Example Suppose that you want to give two movie tickets to your two closest friends. How many ways can you do this? P (4 , 2) ← arrange 2 out of 4 people P (2 , 2) ← divide out order of 2 selected people Combinations A combination of n things taken r at a time is the number of ways to select r things from n distinct things without replacement when the order of selection does not matter. P ( n , r ) ← arrange r out of n items P ( r , r ) ← divide out order of r items

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Generalizing the Concept Example Suppose that you want to give two movie tickets to your two closest friends. How many ways can you do this? P (4 , 2) ← arrange 2 out of 4 people P (2 , 2) ← divide out order of 2 selected people Combinations A combination of n things taken r at a time is the number of ways to select r things from n distinct things without replacement when the order of selection does not matter. P ( n , r ) ← arrange r out of n items P ( r , r ) ← divide out order of r items

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Example Computations Example Find each value 1 C (5 , 0) 2 C (5 , 1) 3 C (5 , 2) 4 C (5 , 3) 5 C (5 , 4) 6 C (5 , 5)

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Example Computations Example Find each value 1 C (5 , 0) 2 C (5 , 1) 3 C (5 , 2) 4 C (5 , 3) 5 C (5 , 4) 6 C (5 , 5)

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Example Computations Example Find each value 5! 5! 1 C (5 , 0) = (5 − 0)!0! = 5!0! = 1 2 C (5 , 1) 3 C (5 , 2) 4 C (5 , 3) 5 C (5 , 4) 6 C (5 , 5)

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Example Computations Example Find each value 5! 5! 1 C (5 , 0) = (5 − 0)!0! = 5!0! = 1 2 C (5 , 1) 3 C (5 , 2) 4 C (5 , 3) 5 C (5 , 4) 6 C (5 , 5)

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Example Computations Example Find each value 5! 5! 1 C (5 , 0) = (5 − 0)!0! = 5!0! = 1 5! 5! 2 C (5 , 1) = (5 − 1)!1! = 4!1! = 5 3 C (5 , 2) 4 C (5 , 3) 5 C (5 , 4) 6 C (5 , 5)

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Example Computations Example Find each value 5! 5! 1 C (5 , 0) = (5 − 0)!0! = 5!0! = 1 5! 5! 2 C (5 , 1) = (5 − 1)!1! = 4!1! = 5 3 C (5 , 2) 4 C (5 , 3) 5 C (5 , 4) 6 C (5 , 5)

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Example Computations Example Find each value 5! 5! 1 C (5 , 0) = (5 − 0)!0! = 5!0! = 1 5! 5! 2 C (5 , 1) = (5 − 1)!1! = 4!1! = 5 5! 5! 3 C (5 , 2) = (5 − 2)!2! = 3!2! = 10 4 C (5 , 3) 5 C (5 , 4) 6 C (5 , 5)

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Example Computations Example Find each value 5! 5! 1 C (5 , 0) = (5 − 0)!0! = 5!0! = 1 5! 5! 2 C (5 , 1) = (5 − 1)!1! = 4!1! = 5 5! 5! 3 C (5 , 2) = (5 − 2)!2! = 3!2! = 10 4 C (5 , 3) 5 C (5 , 4) 6 C (5 , 5)

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Example Computations Example Find each value 5! 5! 1 C (5 , 0) = (5 − 0)!0! = 5!0! = 1 5! 5! 2 C (5 , 1) = (5 − 1)!1! = 4!1! = 5 3 C (5 , 2) = 5! 5! (5 − 2)!2! = 3!2! = 10 5! 5! 4 C (5 , 3) = (5 − 3)!3! = 2!3! = 10 5 C (5 , 4) 6 C (5 , 5)

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Example Computations Example Find each value 5! 5! 1 C (5 , 0) = (5 − 0)!0! = 5!0! = 1 5! 5! 2 C (5 , 1) = (5 − 1)!1! = 4!1! = 5 3 C (5 , 2) = 5! 5! (5 − 2)!2! = 3!2! = 10 5! 5! 4 C (5 , 3) = (5 − 3)!3! = 2!3! = 10 5 C (5 , 4) 6 C (5 , 5)

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Example Computations Example Find each value 5! 5! 1 C (5 , 0) = (5 − 0)!0! = 5!0! = 1 2 C (5 , 1) = 5! 5! (5 − 1)!1! = 4!1! = 5 5! 5! 3 C (5 , 2) = (5 − 2)!2! = 3!2! = 10 5! 5! 4 C (5 , 3) = (5 − 3)!3! = 2!3! = 10 5! 5! 5 C (5 , 4) = (5 − 4)!4! = 1!4! = 5 6 C (5 , 5)

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Example Computations Example Find each value 5! 5! 1 C (5 , 0) = (5 − 0)!0! = 5!0! = 1 2 C (5 , 1) = 5! 5! (5 − 1)!1! = 4!1! = 5 5! 5! 3 C (5 , 2) = (5 − 2)!2! = 3!2! = 10 5! 5! 4 C (5 , 3) = (5 − 3)!3! = 2!3! = 10 5! 5! 5 C (5 , 4) = (5 − 4)!4! = 1!4! = 5 6 C (5 , 5)

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Example Computations Example Find each value 5! 5! 1 C (5 , 0) = (5 − 0)!0! = 5!0! = 1 2 C (5 , 1) = 5! 5! (5 − 1)!1! = 4!1! = 5 5! 5! 3 C (5 , 2) = (5 − 2)!2! = 3!2! = 10 5! 5! 4 C (5 , 3) = (5 − 3)!3! = 2!3! = 10 5! 5! 5 C (5 , 4) = (5 − 4)!4! = 1!4! = 5 6 C (5 , 5) = 5! 5! (5 − 5)!5! = 0!5! = 1

Developing Combinations Examples of Combinations Combinations vs. Permutations Conclusion Example Computations Example Find each value 1 C (5 , 0) = 5! 5! (5 − 0)!0! = 5!0! = 1 5! 5! 2 C (5 , 1) = (5 − 1)!1! = 4!1! = 5 5! 5! 3 C (5 , 2) = (5 − 2)!2! = 3!2! = 10 Symmetric! 5! 5! 4 C (5 , 3) = (5 − 3)!3! = 2!3! = 10 5 C (5 , 4) = 5! 5! (5 − 4)!4! = 1!4! = 5 5! 5! 6 C (5 , 5) = (5 − 5)!5! = 0!5! = 1