{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture_4 - Combinations Permutations How many different...

Info icon This preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
1 Combinations & Permutations How many different ways can you chose K objects from N? How many ways can you chose K objects from N?
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2 Learning Objective 4 : Factorials Rules for factorials: n!=n*(n-1)*(n-2)…2*1 1!=1 0!=1 For example, 4!=4*3*2*1=24
Image of page 2
Combinations The number of ways of arranging k successes in a series of n observations (with constant probability p of success) is the number of possible combinations (unordered sequences). This can be calculated with the binomial coefficient : )! ( ! ! k n k n n k C k n - = = The “ n _perm_ k ” uses the factorial notation “ ! ”. The factorial n! for any strictly positive whole number n is: n! = n × ( n − 1) × ( n − 2)×· · ·×3 × 2 × 1 Where k = 0, 1, 2, ....... , or n
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Permutations The number of ways of arranging k successes in a series of n observations (with constant probability p of success) is the number of possible permutations (ordered sequences). This can be calculated with the: )! ( ! k n n p k n - = The binomial coefficient “ n _choose_ k ” uses the factorial notation “ ! ”. The factorial n! for any strictly positive whole number n is: n! = n × ( n − 1) × ( n − 2)×· · ·×3 × 2 × 1 Where k = 0, 1, 2, ....... , or n
Image of page 4
Difference If the order doesn't matter, it is a Combination If the order does matter it is a Permutation . 1. Permutations with Repetition n * n * n…. -Choosing n, r times, allowed to repeat 2. Permutations without Repetition -Choosing n, and not allowed to repeat (Factorial) How many ways can first and second place be awarded to 10 people?
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Difference If the order doesn't matter, it is a Combination If the order does matter it is a Permutation . For example, let us say balls 1, 2 and 3 were chosen. These are the possibilities: Order does matter Order doesn't matter 1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1 1 2 3
Image of page 6
Difference If the order doesn't matter, it is a Combination (No repeats) If the order does matter it is a Permutation . (Repeats) Combination: Picking a team of 3 people from a group of 10. C(10,3) = 10!/(7! * 3!) = 10 * 9 * 8 / (3 * 2 * 1) = 120. Permutation: Picking a President, VP and Waterboy from a group of 10. P(10,3) = 10!/7! = 10 * 9 * 8 = 720. Combination: Choosing 3 desserts from a menu of 10. C(10,3) = 120. Permut: Listing your 3 favorite desserts, in order, from a menu of 10. P(10,3) = 720.
Image of page 7

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Difference If the order doesn't matter, it is a Combination (No repeats) If the order does matter it is a Permutation . (Repeats) Suppose we want to find the number of ways to arrange the three letters in the word CAT in different two-letter groups where CA is different from AC and there are no repeated letters. Because order matters, we're finding the number of permutations of size 2 that can be taken from a set of size 3. This is often written 3_P_2. We can list them as: CA CT AC AT TC TA When we want to find the number of combinations of size 2 without repeated letters that can be made from the three letters in the word CAT, order doesn't matter; AT is the same as TA. We can write out the three combinations of size two that can be taken from this set of size three: CA CT AT We say '3 choose 2' and write 3_C_2
Image of page 8
9 Probability Distributions How Can We Find Probabilities When Each Observation Has Two Possible Outcomes?
Image of page 9

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern