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Day 19

Day 19. Permutation and Combination. Some Leftover. Tree diagram A graphical representation that helps visualizing a multiple step experiment. Information needs to be included: Outcomes for each step; experiment outcome Example: Toss a coin 3 times My weekend plan to Turkey Run State Park.

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Day 19

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  1. Day 19 Permutation and Combination

  2. Some Leftover • Tree diagram • A graphical representation that helps visualizing a multiple step experiment. • Information needs to be included: • Outcomes for each step; experiment outcome • Example: • Toss a coin 3 times • My weekend plan to Turkey Run State Park

  3. Permutation • A permutation of r objects is any ordered arrangement for r distinct objects from a total of m objects. • ** (a, b, c), (a, c, b) and (c, a, b) are three different permutations • Notation: mPr or (m)r. • Formula: mPr = m! / (m-r)! • A special case: mPm =m!

  4. Idea of permutation • Basically, permutation does two things • 1. Select a subset (or the whole set) out of a given set • 2. Put the selected subset in order

  5. Example • There are five people, A, B, C, D and E. You want to line them up but A and B has to stay together. How many possible ways can we arrangement these five people? • We can first put the three people other than A and B in order (3P3), then fill A and B into the four spaces created by C, D and E, (4), finally alternate the order of A and B (2). • The answer is finally 3!*4*2

  6. Example • In how many ways can the letter PIGGY be arranged? • Using the formula on the next page, we have 5!/(1!1!1!2!) • How about we want to put the two G’s together? • This is similar to the first example: 1.put P,I,Y in order (3P3); 2. Fill in the two G’s, (4). Since two G’s are identical, we don’t have to alternate the order, the answer is finally 3!*4

  7. There is a formula for that kind of problem • There are m! / (m1!*m2!*…*mk!) different permutations of m subjects of which m1, m2, …, mr are alike, respectively. • Example • If we have 3 balls, two red and one green, how many possible ways to put them in order? • RRG , RGR, GRR . That is 3! / 2! = 3.

  8. Example • At an academic conference, 12 faculties are going to take a picture together. There are 3 professors, 5 associate professors and 4 assistant professors. How many ways to line them up? • This one can be done using the formula: 12!/(3!4!5!) • At a party after the conference, 12 people want to get something to drink. In the fridge, there are 3 Miller Genuine Drafts, 5 Coors Light and 4 Coke Zeros. How many ways of distributing the drinks? • In this problem, drinks of the same brand are considered identical, therefore, this is a combination problem. 12C39C4

  9. My new bike lock • My new bike lock has three slots numbered between 1 and 48. How many different ways can the code be set if: • 1. No restrictions at all (48^3) • 2. No two consecutive numbers may be the same (48*47*47)

  10. My new bike lock (contd) • 3. None of the numbers may be the same • (48*47*46) • 4. The third number must be lower than the second. • 48*(0+1+2+3+…+47)

  11. Password Problem • You are required to select a 6-character case-sensitive password for an online account. Each character could be upper-case or lower-case letter or a number from 0 to 9. • i. No restrictions (62^6) • ii. The first character can not be a number 52*(62^5)

  12. Password problem contd • iii. The last four characters must all be different (62^3)*61*60*59 • iv. There must be at least one capital letter and at least one number ? • This is a very difficult problem and we will spend time on it later.

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