2' Combinatorial Methods

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2. Combinatorial Methods
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2.1 Introduction

• If the sample space is finite and furthermore
  sample points are all equally likely, then
• P(A)N(A)/N(S)
• So we study combinatorial analysis here,
• which deals with methods of counting.

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2.2 Counting principle

• Ex 2.1 How many outcomes are there if we throw 5
  dice?
• Ex 2.2 In tossing 4 fair dice,
• P(at least one 3 among these 4
  dice)?

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• Ex 2.3 Virginia wants to give her son, Brian, 14
  different baseball cards within a 7-day period.
  If Virginia gives Brian cards no more than once a
  day, in how many way can this be done?
• Ex 2.6 (Standard Birthday Problem)
• P(at least two among n people have the same
  Bday)?

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Counting principle

• Thm 2.3 A set with n elements has 2n subsets.
• Ex 2.9 Mark has 4. He decides to bet 1 on the
  flip of a fair coin 4 times. What is the
  probability that (a) he breaks even (b) he wins
  money?(use tree diagram)

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2.3 Permutations

• Ex 2.10 3 people, Brown, Smith, and Jones, must
  be scheduled for job interviews. In how many
  different orders can this be done?

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• Ex 2.11 2 anthropology, 4 computer science, 3
  statistics, 3 biology, and 5 music books are put
  on a bookshelf with a random arrangement. What
  is the probability that the books of the same
  subject are together?

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Permutations

• Ex 2.12 If 5 boys and 5 girls sit in a row in a
  random order,
• P(no two children of the same
  sex sit together)?

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Permutations

• Thm 2.4 The number of distinguishable
  permutations of n objects of k different types,
  where n1 are alike, n2 are alike, , nk are alike
  and nn1n2nk is

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Permutations

• Ex 2.13 How many different 10-letter codes can be
  made using 3 as, 4 bs, and 3 cs?
• Ex 2.14 In how many ways can we paint 11 offices
  so that 4 of them will be painted green, 3
  yellow, 2 white, and the remaining 2 pink?

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Permutations

• Ex 2.15 A fair coin is flipped 10 times.
  P(exactly 3 heads)?

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2.4 Combinations

• Ex 2.16 In how many ways can 2 math and 3 biology
  books be selected from 8 math and 6 biology
  books?

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Combinations

• Ex 2.17 45 instructors were selected randomly to
  ask whether they are happy with their teaching
  loads. The response of 32 were negative. If
  Drs. Smith, Brown, and Jones were among those
  questioned. P(all 3 gave negative responses)?

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Combinations

• Ex 2.18 In a small town, 11 of the 25
  schoolteachers are against abortion, 8 are for
  abortion, and the rest are indifferent. A random
  sample of 5 schoolteachers is selected for an
  interview. (a)P(all 5 are for abortion)?
  (b)P(all 5 have the same opinion)?

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Combinations

• Ex 2.19 In Marylands lottery, player pick 6
  integers between 1 and 49, order of selection
  being irrelevant.
• P(grand prize)? P(2nd prize)? P(3rd
  prize)?

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Combinations

• Ex 2.20 7 cards are drawn from 52 without
  replacement.
• P(at least one of the cards is a king)?
• Ex 2.21 5 cards are drawn from 52. P(full
  house)?

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Combinations

• Ex 2.22 A professor wrote n letters and sealed
  them in envelopes. P(at least one letter was
  addressed correctly)?
• Hint Let Ei be the event that ith letter
  is addressed correctly. Compute P(E1UUEn) by
  inclusion-exclusion principle.

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Combinations

• Thm 2.5 (Binomial expansion)
• Ex 2.23 What is the coefficient of x2y3 in the
  expansion of (2x3y)5?

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Combinations

• Ex 2.24 Evaluate the sum

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Combinations

• Ex 2.25 Evaluate the sum

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Combinations

• Ex 2.26 Prove that

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Combinations

• Ex 2.27 Prove the inclusion-exclusion principle.

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Combinations

• Ex 2.26 Distribute n distinguishable balls into k
  distinguishable cells so that n1 balls are
  distributed into the first cell, n2 balls into
  the second cell, , nk balls into the kth cell,
  where n1n2nkn. How many possible ways?
• Sol

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Combinations

• Thm 2.6 (Multinomial expansion).

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2.5 Stirlings formula