Further Counting Problems Example 1 A jar contains 1

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Further CountingProblems
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Example 1

• A jar contains 10 marbles 6 are red and 4 are
  blue. A sample of 4 marbles is to be selected.
• a.) How many different samples are possible?
• b.) How many samples contain all red marbles?
• c.) How many samples contain 2 red marbles
• and 2 blue marbles?
• d.) How many samples contain at least 3 red
• marbles?
• e.) How many samples contain a different
• number of red marbles than blue marbles?

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Example 2

• An experiment consists of tossing a coin 10 times
  and observing the sequence of heads and tails.
• a.) How many different outcomes are possible?
• b.) How many different outcomes have exactly 3
• tails?
• c.) How many different outcomes have at least 2
• heads?
• d.) How many different outcomes have 4 tails or
  5
• tails?

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Example 3

• A package contains 100 fuses, of which 10 are
  defective. A sample of 6 fuses is selected at
  random.
• a.) How many different samples are there?
• b.) How many of the samples contain 2
• defective fuses?
• c.) How many of the samples contain at
• least 1 defective fuse?

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Example 4

• The new book shelf at a library contains three
  novels, eight biographies, and six how-to books.
  In how many ways can the books be arranged on the
  shelf if the books for each category are placed
  together?

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Example 5

• A committee has three male and four female
  members. In how many ways can a subcommittee
  consisting of two females and 1 male be selected?

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Example 6

• A football team plays 12 games one season. In
  how many ways can these games result in nine
  wins, three losses and no ties?

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Example 7

• During practice, a basketball player shoots 15
  free-throws. In how many ways can the outcomes
  result in 10 hits and 5 misses?

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Example 8

• The shortest word containing all five vowels is
  SEQUOIA. How many seven-letter sequences of
  letters, with no repeated letters, contain all
  five vowels?

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Example 9

• A truck driver has to deliver bread to five
  grocery stores and four convenience stores. In
  how many different ways can he schedule the order
  of his stops?

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Example 10

• Of the 18 new programmers hired by a major
  software company, 5 will be selected to work on
  programming languages, 4 will be selected to work
  on word processing software, and 7 will be
  selected to work on spreadsheet software. In how
  many ways can this be done?

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Example 11

• How many 5-card poker hands consist of exactly 3
  tens?

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Example 12

• How many poker hands consist of 3 cards of one
  denomination and 2 cards of another denomination?
  (In other words, how many poker hands contain a
  full house?)

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Example 13

• A dance team knows 15 routines of which 7 are
  tap, 6 are ballet, and 2 are modern. The program
  can consist of any five routines. In how many
  ways can the manager choose which routines to
  present if there are no restrictions?

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Example 14

• In how many ways can 5 married couples sit in a
  row if men and women alternate?

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Example 15

• The dean at a small college wishes to form an
  experimental section of General Psychology with
  25 students chosen from 22 women and 20 men
  interested in social science.
• a.) How many different such classes are
• possible?
• b.) The dean decides the class should have 13
• women and 12 men. How many different
• classes are possible now?